Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 59, pp. 1–27. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONNEGATIVE CONTROLLABILITY FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC EQUATIONS WITH APPLICATION TO CLIMATE SCIENCE GIUSEPPE FLORIDIA Communicated by Jesus Ildefonso Diaz Abstract. We consider a nonlinear degenerate reaction-diffusion equation. First we prove that if the initial state is nonnegative, then the solution re- mains nonnegative for all time. Then we prove the approximate controllability between nonnegative states via multiplicative controls, this is done using the reaction coefficient as control. 1. Introduction In this article, we study the one-dimensional semilinear reaction-diffusion equa- tion ut − (a(x)ux)x = α(x, t)u+ f(x, t, u), (x, t) ∈ QT := (−1, 1)× (0, T ), T > 0, where a ∈ C([−1, 1]) ∩ C1(−1, 1) is strictly positive on (−1, 1) and a(±1) = 0 (for example a(x) = (1 − x2)η with η > 0), α is a bounded function on QT and f(·, ·, u) is a suitable non-linearity that will be defined below. The above semilinear equation is a degenerate parabolic equation since the diffusion coefficient vanishes at the boundary points of [−1, 1]. Our interest in this degenerate reaction-diffusion equations is motivated by its applications to the energy balance models in climate science. For example, the Budyko-Sellers model that is obtained from the above equation when a(x) = 1−x2. We devote the entire Section 4 to the presentation of applications of degenerate equations to climate science. In our mathematical study we need to distinguish two classes of degenerate problems: weakly degenerate problems (WDeg) (see [14, 36]) when the degenerate diffusion coefficient is such that 1 a ∈ L 1(−1, 1) (e.g. a(x) = √ 1− x2), and strongly degenerate problems (SDeg) (see [13, 35]) when 1 a 6∈ L 1(−1, 1) (if a ∈ C1([−1, 1]) follows that 1 a 6∈ L 1(−1, 1), e.g. a(x) = 1 − x2). It is well-known [1] that, in the (WDeg) case, all functions in the domain of the corresponding differential operator 2010 Mathematics Subject Classification. 93C20, 35K10, 35K65, 35K57, 35K58. Key words and phrases. Semilinear degenerate reaction-diffusion equations; energy balance models in climate science; approximate controllability; multiplicative controls; nonnegative states. c©2020 Texas State University. Submitted March 10, 2020. Published June 15, 2020. 1 2 G. FLORIDIA EJDE-2020/59 possess a trace on the boundary, in spite of the fact that the operator degener- ates at such points. Thus, in the (WDeg) case we can consider the general Robin type boundary conditions, in a similar way to the uniformly parabolic case. Con- versely, in the harder (SDeg) case, one is limited to only the weighted Neumann type boundary conditions. This preamble allows us to justify the following general problem formulation. 1.1. Problem formulation. Let us introduce the semilinear degenerate parabolic Cauchy problems: ut − (a(x)ux)x = α(x, t)u+ f(x, t, u) in QT := (−1, 1)× (0, T ) { β0u(−1, t) + β1a(−1)ux(−1, t) = 0 t ∈ (0, T ) γ0 u(1, t) + γ1 a(1)ux(1, t) = 0 t ∈ (0, T ) for (WDeg) a(x)ux(x, t)|x=±1 = 0 for (SDeg) u(x, 0) = u0(x) ∈ L2(−1, 1), (1.1) where the reaction coefficient α(x, t) ∈ L∞(QT ) will represent the multiplicative control (that is the variable function through which we can act on the system), and α is chosen, in this article, as a piecewise static function (in the sense of Definition 1.1). We consider problem (1.1) under the following assumptions: (A1) The function f : QT × R→ R satisfies: • (x, t, u) 7→ f(x, t, u) is a Carathéodory function on QT × R, that is ? (x, t) 7→ f(x, t, u) is measurable, for every u ∈ R, ? u 7→ f(x, t, u) is a continuous function, for a.e. (x, t) ∈ QT ; • t 7→ f(x, t, u) is locally absolutely continuous for a.e. x ∈ (−1, 1), for every u ∈ R, and ft(x, t, u)u ≥ −νu2 for a.e. t ∈ (0, T ); • there exist constants δ∗ ≥ 0, ϑ ∈ [1, ϑsup), ϑsup ∈ {3, 4}, and ν ≥ 0 such that for a.e. (x, t) ∈ QT and all u, v ∈ R, we have |f(x, t, u)| ≤ δ∗ |u|ϑ, (1.2) −ν ( 1 + |u|ϑ−1 + |v|ϑ−1 ) (u− v)2 ≤ ( f(x, t, u)− f(x, t, v) ) (u− v) ≤ ν(u− v)2, (1.3) (A2) The function a ∈ C([−1, 1]) ∩ C1(−1, 1) satisfies a(x) > 0, ∀x ∈ (−1, 1), a(−1) = a(1) = 0 . Then, we consider the following two cases: • case (WDeg): if 1 a ∈ L 1(−1, 1), then ϑsup = 4. In (1.1) we consider the Robin boundary conditions, where β0, β1, γ0, γ1 ∈ R, with β2 0 +β2 1 > 0 and γ20 + γ21 > 0 satisfy the sign condition: β0β1 ≤ 0 and γ0γ1 ≥ 0; • case (SDeg): if 1 a 6∈ L1(−1, 1) and ξa(x) := ∫ x 0 1 a(s)ds ∈ Lqϑ(−1, 1), where qϑ = max { 1+ϑ 3−ϑ , 2ϑ − 1 } , then ϑsup = 3. In (1.1) we consider the weighted Neumann boundary conditions. To clarify the kind of multiplicative controls used, we recall the definition of piecewise static function. EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 3 Definition 1.1. We say that a function α ∈ L∞(QT ) is piecewise static (or a simple function with respect to the variable t), if there exist m ∈ N, αk(x) ∈ L∞(−1, 1) and tk ∈ [0, T ], tk−1 < tk, k = 1, . . . ,m with t0 = 0 and tm = T , such that α(x, t) = α1(x)χ[t0,t1](t) + m∑ k=2 αk(x)χ(tk−1,tk](t), (1.4) where χ[t0,t1] and χ(tk−1,tk] are the indicator function of [t0, t1] and (tk−1, tk], re- spectively. Sometimes, for clarity purposes, we will call the function α in (1.4) an m-step piecewise static function 1.2. Main results. In this article, we study the controllability of (1.1) using mul- tiplicative controls, that is the reaction coefficients α(x, t). First, we find that the following general nonnegative result holds also for the degenerate PDE of system (1.1). That is, if the initial state is nonnegative the corresponding strong solution to (1.1) remains nonnegative for all time. For the notion of strict/strong solutions of the nonlinear degenerate problem (1.1) see Section 2. The following result is classic only for the uniformly parabolic (non degenerate) case. Proposition 1.2. Let u0 ∈ L2(−1, 1) such that u0(x) ≥ 0 a.e. x ∈ (−1, 1). Let u be the corresponding unique strong solution of (1.1). Then u(x, t) ≥ 0, for a.e. (x, t) ∈ QT . A consequence of this result, the solution to (1.1) cannot be steered from a nonnegative initial state to any target state which is negative on a nonzero measure set in the space domain, regardless of the choice of the reaction coefficient α(x, t) as multiplicative control. In Theorem 1.4 below, we obtain an optimal goal, that is, we approximately control system (1.1) between nonnegative states via multiplicative controls at any time. Definition 1.3. System (1.1) is said to be non-negatively globally approximately controllable in L2(−1, 1) at any time T > 0, by means of multiplicative controls α, if for any nonnegative u0, u ∗ ∈ L2(−1, 1) with u0 6= 0, and for every ε > 0 there exists a piecewise static multiplicative control α = α(ε, u0, u ∗) in L∞(QT ), such that for the corresponding strong solution u(x, t) of (1.1) we have ‖u(·, T )− u∗‖L2(−1,1) < ε. Now we can state the main controllability result. Theorem 1.4. The nonlinear degenerate system (1.1) is non-negatively globally approximately controllable in L2(−1, 1) at any time T > 0, by means of 2-steps piecewise static multiplicative controls. The outline of this article is as follows: The proofs of the main results are given in Section 3. In Section 2 we recall the well-posedness of (1.1), in particular we introduce the notions of strict and strong solutions, and we give some useful estimates and properties for this kind of degenerate PDEs, that we use in Section 3. The proofs of the existence and uniqueness results for strict and strong solutions are contained in Section 5. In Section 4 we present some motivations for studying degenerate parabolic problems with the above structure, in particular we introduce the Budyko-Sellers model, an energy balance model in climate science. 4 G. FLORIDIA EJDE-2020/59 1.3. State of the art in multiplicative controllability and degenerate par- abolic equations. Control theory appeared in the second part of the previous century in the context of linear ordinary differential equations and was motivated by several engineering, Life sciences and economics applications. Then, it was ex- tended to various linear partial differential equations (PDEs) governed by additive locally distributed controls (see [1, 3, 19, 24, 32, 34, 47]), or by boundary con- trols. Methodologically-speaking, these kind of controllability results for PDEs are typically obtained using the linear duality pairing technique between the control- to-state mapping at hand and its dual observation map (see the Hilbert Uniqueness Method - HUM - introduced in 1988 by J. L. Lions), sometimes using the Carleman estimates tool (see [1, 15, 17]). If the above map is nonlinear, as it happens in our case for the multiplicative controllability, in general the aforementioned approach does not apply. From the point of view of applications, the approach based on multiplicative controls seems more realistic than the other kinds of controllability, since additive and boundary controls do not model in a realistic way the problems that involve inputs with high energy levels; such as energy balance models in climate science (see Section 4), chemical reactions controlled by catalysts, nuclear chain reactions, smart materials, social science, ecological population dynamic (see [49]) and biomedical models. An important class of biomedical reaction-diffusion problems consists in the models of tumor growth (see, e.g. Section 7 “Control problems” of the survey paper [7] by Bellomo and Preziosi). As regards degenerate reaction-diffusion equations there are also interesting models in population genetics, in particular we recall the Fleming-Viot model (see Epstein’s and Mazzeo’s book [30]). The above considerations motivate our investigation of the multiplicative con- trollability. As regards the topic of multiplicative controllability of PDEs we recall the pioneering paper [5] by Ball, Marsden, and Slemrod, and in the framework of the Schrödinger equation we especially mention [6] by Beauchard and Laurent, and [22] by Coron, Gagnon and Moranceu. As regards parabolic and hyperbolic equations we focus on some results by Khapalov contained in the book [41], and in the references therein. The main results of this paper deal with approximate multiplicative controllabil- ity of semilinear degenerate reaction-diffusion equations. This study is motivated by its applications (see in Section 4 to an energy balance model in climate sci- ence: the Budyko-Sellers model) and also by the classical results that hold for the corresponding non-degenerate reaction-diffusion equations, governed via the coeffi- cient of the reaction term (multiplicative control). For the above class of uniformly parabolic equations there are some important obstructions to multiplicative con- trollability due to the strong maximum principle (see the seminal papers by Diaz [25] and [26], and also the papers [16, 41]), this implies the well-known nonneg- ative constraint. In Proposition 1.2 we extend the nonnegative constraint to the semilinear degenerate system (1.1). This motivates our investigations regarding the nonnegative controllability for the semilinear degenerate system (1.1) with general weighted Robin/Neumann boundary conditions. Regarding the nonnegative controllability for reaction-diffusion equations, first, Khapalov in [41] obtained the nonnegative approximate controllability in large time of the one dimensional heat equation via multiplicative controls. Thus, the author and Cannarsa considered the linear degenerate problem associated with (1.1), both EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 5 in the weakly degenerate (WDeg) case, in [14], and in the strongly degenerate (SDeg) case, in [13]. Then, the author in [35] investigated semilinear strongly de- generate problems. This article can be seen as the final step of the study started in [13, 14, 35], where the global nonnegative approximate controllability was obtained in large time. Indeed, in this article we introduce a new proof, that permits us to obtain the nonnegative controllability in arbitrary small time and consequently at any time, instead of large time. Moreover, the proof, contained in [35], of the nonnegative controllability in large time for the (SDeg) case has the further ob- struction that permitted to treat only superlinear growth, with respect to u, of the nonlinearity function f(x, t, u). While the new proof, adopted in this paper, permits us to control also linear growth of f , with respect to u. Finally, we mention some recent papers about the approximate multiplicative controllability for reaction-diffusion equations between sign-changing states: in [16] by the author with Cannarsa and Khapalov regarding a semilinear uniformly para- bolic system, and [37] by the author with Nitsch and Trombetti, concerning degen- erate parabolic equations. Furthermore, some interesting contributions about exact controllability issues for evolution equations via bilinear controls have recently ap- peared, in particular we mention [2] by Alabau-Boussouira, Cannarsa and Urbani, and [29] by Duprez and Lissy. To complete the discussion regarding the multiplicative controllability, we note that recently there has been an increasing interest in these topics; many authors are starting to extend the above results from reaction-diffusion equations to other operators. In [50], Vancostenoble proved a nonnegative controllability result in large time for a linear parabolic equation with singular potential, following the approach of [13] and [14]. An interesting work in progress, using the technique of this paper, consists of approaching the problem of the approximate controllability via multiplicative control of nonlocal operators, e.g. the fractional heat equation studied in [10] by Biccari, Warma and Zuazua. Other interesting open problems are suggested in [38, 42]. 2. Well-posedness The well-posedness of case (SDeg) in problem (1.1) was introduced in [35], while the well-posedness of case (WDeg) in problem (1.1) was presented in [36]. To study the well-posedness of (1.1), it is necessary to introduce in the weighted Sobolev spaces H1 a(−1, 1) and H2 a(−1, 1), and their main propertie. Also, in Section 2.2 we introduce the notions of strict and strong solutions semilinear degenerate problems, and we give the corresponding existence and uniqueness results, that are proved in Section 5. 2.1. Weighted Sobolev spaces. Let a ∈ C([−1, 1]) ∩ C1(−1, 1) such that the assumption (A2) holds, we define the following spaces: H1 a(−1, 1) = { {u ∈ L2(−1, 1) ∩AC([−1, 1]) : √ a ux ∈ L2(−1, 1)} for (WDeg) {u ∈ L2(−1, 1) ∩ACloc(−1, 1) : √ a ux ∈ L2(−1, 1)} for (SDeg), where AC([−1, 1]) denotes the space of the absolutely continuous functions on [−1, 1], and ACloc(−1, 1) denotes the space of the locally absolutely continuous functions on (−1, 1). H2 a(−1, 1) := {u ∈ H1 a(−1, 1) : aux ∈ H1(−1, 1)}. 6 G. FLORIDIA EJDE-2020/59 See [1, 13, 14, 35, 36] for the main properties of the weighted Sobolev spaces. In particular we note that H1 a(−1, 1) and H2 a(−1, 1) are Hilbert spaces with their natural scalar products that induce, respectively, the norms: ‖u‖21,a := ‖u‖2L2(−1,1) + |u|21,a, ‖u‖22,a := ‖u‖21,a + ‖(aux)x‖2L2(−1,1), where |u|21,a := ‖ √ aux‖2L2(−1,1) is a seminorm. Remark 2.1. The space H1 a(−1, 1) is embedded in L∞(−1, 1) only in the weakly degenerate case (see [1, 13, 14, 35]). The following proposition was presented in [18, Proposition 2.1], [35, Appendix], and [12, Lemma 2.5]. Proposition 2.2. In the (SDeg) case, for every u ∈ H2 a(−1, 1) we have lim x→±1 a(x)ux(x) = 0 and au ∈ H1 0 (−1, 1) . Some spectral properties. Let us define the operator (A0, D(A0)) by D(A0) =  { u ∈ H2 a(−1, 1) : { β0u(−1) + β1a(−1)ux(−1) = 0 γ0u(1) + γ1a(1)ux(1) = 0 } for (WDeg) H2 a(−1, 1) for (SDeg) A0u = (aux)x , ∀u ∈ D(A0). (2.1) Remark 2.3. We note that in the (SDeg) case, for every u ∈ D(A0) Proposition 2.2 guarantees that u satisfies the weighted Neumann boundary conditions. When α ∈ L∞(−1, 1), we define the operator (A,D(A)) as D(A) = D(A0) Au = (aux)x + αu, ∀u ∈ D(A). (2.2) We recall some spectral results obtained from [14] for (WDeg), and from [13] for (SDeg). Proposition 2.4. (A,D(A0)) is a closed, self-adjoint, dissipative operator with dense domain in L2(−1, 1). Therefore, A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators on L2(−1, 1). Proposition 2.4 allows us to obtain the following result. Proposition 2.5. There exists an increasing sequence {λp}p∈N, with λp → +∞ as p → ∞, such that −λp are the eigenvalues of (A0, D(A0)), and the corresponding eigenfunctions {ωp}p∈N form a complete orthonormal system in L2(−1, 1). Remark 2.6. When a(x) = 1 − x2, that is in the case of the Budyko-Sellers model, the orthonormal eigenfunctions of (A0, D(A0)) are reduced to Legendre’s polynomials (see [45, Section 5.6] and [35, Remark 3.2]). EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 7 Spaces involving time: B(QT ) and H(QT ). Given T > 0, let us define the Banach spaces: B(QT ) := C([0, T ];L2(−1, 1)) ∩ L2(0, T ;H1 a(−1, 1)) with the norm ‖u‖2B(QT ) = sup t∈[0,T ] ‖u(·, t)‖2L2(−1,1) + 2 ∫ T 0 ∫ 1 −1 a(x)u2xdx dt , and H(QT ) := L2(0, T ;D(A0)) ∩H1(0, T ;L2(−1, 1)) ∩ C([0, T ];H1 a(−1, 1)) with the norm ‖u‖2H(QT ) = sup [0,T ] ( ‖u‖2L2(−1,1) + ‖ √ aux‖2L2(−1,1) ) + ∫ T 0 ( ‖ut‖2L2(−1,1) + ‖(aux)x‖2L2(−1,1) ) dt. The following embedding lemma was obtained for the space H(QT ): in [35] for the (SDeg) case, and in [36] for the (WDeg) case. Lemma 2.7. Let ϑ ≥ 1. Then H(QT ) ⊂ L2ϑ(QT ) and ‖u‖L2ϑ(QT ) ≤ cT 1 2ϑ ‖u‖H(QT ), where c is a positive constant. We note that Lemma 2.7 holds in a more general setting than under the assump- tions (A1) and (A2), where ϑ ∈ [1, ϑsup), with ϑsup ∈ {3, 4}. 2.2. Existence and uniqueness of solutions of semilinear degenerate prob- lems. To study the well-posedness, we represent the semilinear problem (1.1) using the following abstract setting in the Hilbert space L2(−1, 1), u′(t) = (A0 + α(t)I)u(t) + φ(u) , t > 0 u(0) = u0 ∈ L2(−1, 1) , (2.3) where A0 is the operator defined in (2.1), I is the identity operator and, for every u ∈ B(QT ), the Nemytskii operator associated with the problem (1.1) is defined as φ(u)(x, t) := f(x, t, u(x, t)), ∀(x, t) ∈ QT . (2.4) The following proposition was proved in [35], for the (SDeg) case, and in [36], for the (WDeg) case. Proposition 2.8. Let 1 ≤ ϑ < ϑsup, let f : QT ×R→ R and assume that (A1) and (A2) hold. Then φ : B(QT ) → L1+ 1 ϑ (QT ), defined in (2.4), is a locally Lipschitz continuous map and φ(H(QT )) ⊆ L2(QT ). The above proposition justifies the introduction of the notions of strict solutions and strong solutions. Such notions are classical in PDE theory, see, for instance, the book [8, pp. 62-64] (see also [35, 36]) and the pioneer paper [39] by Friedrichs. 8 G. FLORIDIA EJDE-2020/59 Strict solutions. In this section we give the notion of solutions of (1.1) with initial state in H1 a(−1, 1), introduced in [35] for (SDeg) and in [36] for (WDeg). Definition 2.9. If u0 ∈ H1 a(−1, 1), then u is a strict solution of (1.1), if u ∈ H(QT ) and ut − (a(x)ux)x = α(x, t)u+ f(x, t, u) a.e. in QT := (−1, 1)× (0, T ) { β0u(−1, t) + β1a(−1)ux(−1, t) = 0 a.e. t ∈ (0, T ) γ0u(1, t) + γ1a(1)ux(1, t) = 0 a.e. t ∈ (0, T ), for (WDeg) a(x)ux(x, t) ∣∣ x=±1 = 0 a.e. t ∈ (0, T ) for (SDeg) u(x, 0) = u0(x) x ∈ (−1, 1) , for almost every t in (0, T ). Remark 2.10. Since a strict solution u belongs to H(QT ) ⊆ L2(0, T ;D(A0)), we have u(·, t) ∈ D(A0), for a.e. t ∈ (0, T ). Thus, thanks to the definition of the operator (A,D(A)) given in (2.2) and Remark 2.3, we deduce that the associated boundary conditions hold, for almost every t ∈ (0, T ). The following existence and uniqueness result for strict solutions is proved in Section 5 (see also [35, Appendix B] for (SDeg), and [36] for (WDeg)). Theorem 2.11. For each u0 ∈ H1 a(−1, 1) there exists a unique strict solution u ∈ H(QT ) to (1.1). Strong solutions. In this subsection we introduce the notion of solutions when the initial state belongs to L2(−1, 1). These solutions are called strong solutions and are defined by approximation sequence of strict solutions. Definition 2.12. Let u0 ∈ L2(−1, 1). We say that u ∈ B(QT ) is a strong solution of (1.1), if u(·, 0) = u0 and there exists a sequence {uk}k∈N in H(QT ) such that, as k → ∞, uk → u in B(QT ) and, for every k ∈ N, uk is the strict solution of the Cauchy problem ukt − (a(x)ukx)x = α(x, t)uk + f(x, t, uk) a.e. in QT := (−1, 1)× (0, T ) { β0uk(−1, t) + β1a(−1)ukx(−1, t) = 0 a.e. t ∈ (0, T ) γ0uk(1, t) + γ1a(1)ukx(1, t) = 0 a.e. t ∈ (0, T ) for (WDeg) a(x)ukx(x, t)|x=±1 = 0 a.e. t ∈ (0, T ) for (SDeg) u(x, 0) = u0(x) x ∈ (−1, 1) with initial datum uk(x, 0). Remark 2.13. Let us consider the sequence of strict solutions in Definition 2.12, {uk}k∈N ⊆ H(QT ) such that, as k → ∞, uk → u in B(QT ). Thus, it follows that uk(·, 0)→ u0 in L2(−1, 1), because of the definition of the B(QT )−norm. The next proposition, obtained in [35] for (SDeg) and in [36] for (WDeg), will be very useful in the proof of the main results. EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 9 Proposition 2.14. Let α ∈ L∞(QT ) a piecewise static function and let u0, v0 ∈ L2(−1, 1). Let u, v be the corresponding strong solutions of (1.1), with initial data u0, v0 respectively. Then, ‖u− v‖B(QT ) ≤ CT ‖u0 − v0‖L2(−1,1), (2.5) where CT = e(ν+‖α +‖∞)T and α+ := max{α, 0}. From Proposition 2.14 trivially we have a corollary. Corollary 2.15. Let u0 ∈ L2(−1, 1), α ∈ L∞(QT ), α be a piecewise static function with α(x, t) ≤ 0 in QT , and u be the corresponding strong solution of (1.1). If T ∈ (0, 1/(4ν)), then ‖u‖C([0,T ],L2(−1,1)) ≤ √ 2 ‖u0‖L2(−1,1). (2.6) The following existence and uniqueness result for strong solutions, was given in [35] for (SDeg) and in [36] for (WDeg)). It will be proved in Section 5. Theorem 2.16. For each u0 ∈ L2(−1, 1) and each piecewise static function α ∈ L∞(QT ), there exists a unique strong solution to (1.1). Further estimates. First, we recall the following Lemma that was obtained in [35, Lemma B.2] for the (SDeg) case, and in [36], for the (WDeg) case. In the case of static reaction α ∈ L∞(−1, 1) (a similar argument to that used in Subsection 5.1.4 to prove Theorem 2.11 which permits to extend the following lemma to the case of α ∈ L∞(QT ) piecewise static function). Lemma 2.17. Let α ∈ L∞(QT ) be a piecewise static function and u0 ∈ H1 a(−1, 1). Then the strict solution u ∈ H(QT ) of system (1.1), under the assumptions (A1) and (A2), satisfies ‖u‖H(QT ) ≤ ce kT ‖u0‖1,a, where c = c(‖u0‖1,a) and k are positive constants. Using Lemma 2.7, Proposition 2.8 and Lemma 2.17 we obtain the following Proposition. Proposition 2.18. Let α ∈ L∞(QT ) be a piecewise static function and u0 ∈ H1 a(−1, 1). Let u ∈ H(QT ) be the strict solution of (1.1), under the assumptions (A1) and (A2). Then, the function (x, t) 7→ f(x, t, u(x, t)) belongs to L2(QT ) and ‖f(·, ·, u)‖L2(QT ) ≤ Ce kϑT √ T‖u0‖ϑ1,a , where C = C(‖u0‖1,a) and k are positive constants. Proof. Applying (1.2), Lemma 2.7 and Lemma 2.17, we obtain∫ T 0 ∫ 1 −1 f2(x, t, u) dx dt ≤ δ2∗ ∫ T 0 ∫ 1 −1 |u|2ϑ dx dt ≤ cT‖u‖2ϑH(QT ) ≤ Ce2kϑTT‖u0‖2ϑ1,a, where c and C = C(‖u0‖1,a) and k are positive constants. � 10 G. FLORIDIA EJDE-2020/59 3. Proof of main results In Proposition 1.2 we showed that the solution to (1.1) remains nonnegative for all time when the initial state is nonnegative, regardless of the choice of the mul- tiplicative control α(x, t). In Theorem 1.4 we showed that the global approximate multiplicative controllability between nonnegative states at any time. For brevity of notation, we will use ‖ · ‖, ‖ · ‖∞ and 〈·, ·〉 instead of the norms ‖ · ‖L2(−1,1) and ‖ · ‖L∞(QT ), and the inner product 〈·, ·〉L2(−1,1), respectively. 3.1. Nonnegative solutions. Before proving Proposition 1.2 we give a regularity property of the positive and negative part of a function, that will be used in that proof. For u : (−1, 1) → R, we consider the positive and negative part functions, respectively, u+(x) := max {u(x), 0} , u−(x) := max {0,−u(x)} , x ∈ (−1, 1) . Then u = u+ − u−. We have the following regularity result in weighted Sobolev spaces, obtained as trivial consequence of a classical result for the usual Sobolev spaces, that we can find in [43, Appendix A]. Proposition 3.1. Let u ∈ H1 a(−1, 1), then u+, u− ∈ H1 a(−1, 1). Moreover, (u+)x = { ux(x) if u(x) > 0 0 if u(x) ≤ 0 and (u−)x = { −ux(x) if u(x) < 0 0 if u(x) ≥ 0 . Proof of Proposition 1.2. Case 1: u0 ∈ H1 a(−1, 1). Firstly, we prove Proposition 1.2 under the further assumption that u0 ∈ H1 a(−1, 1). Note that the corresponding unique solution u(x, t) is a strict solution, that is u ∈ H(QT ) = L2(0, T ;D(A0)) ∩H1(0, T ;L2(−1, 1)) ∩ C([0, T ];H1 a(−1, 1)). We denote with u+ and u− the positive and negative part of u, respectively. Since u = u+ − u−, it sufficies to prove that u−(x, t) = 0, a.e. in QT . Multiplying by u− both sides of the equation in (1.1) and integrating on (−1, 1) we obtain∫ 1 −1 utu −dx = ∫ 1 −1 [(a(x)ux)xu − + αuu− + f(x, t, u)u−]dx. (3.1) We start by estimating of the second term in (3.1). Integrating by parts, recalling that u−(·, t) ∈ H1 a(−1, 1) for every t ∈ (0, T ),and using Proposition 3.1 we deduce∫ 1 −1 (a(x)ux)xu − dx = [a(x)uxu −]1−1 − ∫ 1 −1 a(x)ux(u−)x dx = [a(x)uxu −]1−1 + ∫ 1 −1 a(x)u2x dx . (3.2) If β1γ1 6= 0, keeping in mind the boundary conditions, for t ∈ (0, T ) we have [a(x)uxu −]1−1 = a(1)ux(1, t)u−(1, t)− a(−1)ux(−1, t)u−(−1, t) = −γ0 γ1 (u+(1, t)− u−(1, t))u−(1, t) + β0 β1 (u+(−1, t)− u−(−1, t))u−(−1, t) = γ0 γ1 (u−(1, t))2 − β0 β1 (u−(−1, t))2 ≥ 0. (3.3) EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 11 Thus, including the simple case β1γ1 = 0, from (3.2) and (3.3) we obtain∫ 1 −1 (a(x)ux)xu − dx ≥ 0. (3.4) We also have the equality∫ 1 −1 αuu−dx = − ∫ 1 −1 α(u−)2dx, (3.5) moreover, using (1.3) we have∫ 1 −1 f(x, t, u)u− dx = ∫ 1 −1 f(x, t, u+ − u−)u− dx = ∫ 1 −1 f(x, t,−u−)u− dx = − ∫ 1 −1 f(x, t,−u−)(−u−) dx ≥ − ∫ 1 −1 ν(−u−)2 dx = − ∫ 1 −1 ν(u−)2 dx . (3.6) We can compute the first term in (3.1) the following way∫ 1 −1 utu −dx = ∫ 1 −1 (u+ − u−)tu −dx = − ∫ 1 −1 (u−)tu −dx = −1 2 d dt ∫ (u−)2dx . Applying to (3.1) the above equality and (3.4)-(3.6), we have 1 2 d dt ∫ 1 −1 (u−)2dx ≤ ∫ 1 −1 (α(x, t) + ν) (u−)2dx ≤ (‖α‖∞ + ν) ∫ 1 −1 (u−)2dx, so by Gronwall’s Lemma, since u−0 (x) ≡ 0, we obtain∫ 1 −1 (u−(x, t))2dx ≤ e2(ν+‖α‖∞)T ∫ 1 −1 (u−(x, 0))2dx = 0, ∀t ∈ (0, T ). Therefore, u−(x, t) = 0 ∀(x, t) ∈ QT , (3.7) that proves Proposition 1.2 in the case u0 ∈ H1 a(−1, 1). Case 2: u0 ∈ L2(−1, 1). If u0 ∈ L2(−1, 1), u0 ≥ 0 a.e. x ∈ (−1, 1), then there exists {u0k}k∈N ⊆ C∞([−1, 1]), such that u0k ≥ 0 on (−1, 1) for every k ∈ N, and u0k → u0 in L2(−1, 1), as k → ∞. For every k ∈ N, we consider uk ∈ H(QT ) the strict solution to (1.1) with initial state u0k. For the well-posedness there exists u ∈ B(QT ) such that uk → u in B(QT ), as k →∞. This convergence implies that there exists {ukp}p∈N ⊆ {uk}k∈N such that, as p→∞, ukp(x, t)→ u(x, t), a.e. (x, t) ∈ QT . (3.8) For every p ∈ N, we can apply the Case 1 to the system (1.1) with initial datum u0kp , so by (3.7) we deduce ukp(x, t) ≥ 0, a.e. (x, t) ∈ QT , thus from the convergence (3.8) it follows that u(x, t) ≥ 0, a.e. (x, t) ∈ QT . � 12 G. FLORIDIA EJDE-2020/59 3.2. Nonnegative controllability. Proof of Theorem 1.4. Let us fix ε > 0. Since u0, u ∗ ∈ L2(−1, 1), there exist uε0, u ∗ ε ∈ C1([−1, 1]) such that uε0, u ∗ ε > 0 on [−1, 1], ‖u∗ε − u∗‖ < ε 4 , ‖uε0 − u0‖ < √ 2 36Sεeν ε , (3.9) where ν is the nonnegative constant in assumption (A1) and Sε := max x∈[−1,1] {u∗ε(x) uε0(x) } + 1 . (3.10) From (3.9) and (3.10) it follows that there exists η∗ > 0 such that η∗ ≤ u∗ε(x) Sεuε0(x) ≤ 1, ∀x ∈ [−1, 1] . (3.11) The strategy of the proof consists of using two control actions: in the first step we steer the system from the initial state u0 to the intermediate state Sεu ε 0, then in the second step we drive the system from this to u∗ε. In the second step, condition (3.11) will be crucial, that is justify the choice of the intermediate state Sεu ε 0. Step 1: Steering the system from u0 to Sεu ε 0. Let us choose the positive constant bilinear control α(x, t) = α1 := logSε T1 > 0, (x, t) ∈ (−1, 1)× (0, T1), for some T1 > 0. Let us denote by uε(x, t) and u(x, t) the strict and strong solution of (1.1) with initial state uε0 and u0, respectively. So, keeping in mind the abstract formulation (2.3) for problem (1.1), the Duhamel’s principle and the Proposition (2.8), the strict solution uε(x, t) is given in terms of a Fourier series approach, by uε(x, T1) = eα1T1 ∞∑ p=1 e−λpT1〈uε0, ωp〉ωp(x) +Rε(x, T1) , (3.12) with Rε(x, T1) := ∞∑ p=1 [ ∫ T1 0 e(α1−λp)(T1−t)〈f(·, t, uε(·, t)), ωp〉dt ] ωp(x), where {−λp}p∈N are the eigenvalues of the operator (A0, D(A0)), defined in (2.1), and {ωp}p∈N are the corresponding eigenfunctions, that form a complete orthonor- mal system in L2(−1, 1), see Proposition 2.5. We recall that the eigenvalues of the operator (A,D(A)), with Au = A0u+α1u (defined in (2.2)) are obtained from the eigenvalues of the operator (A0, D(A0)) by shift, that is we have {−λp + α1}p∈N, and the corresponding orthonormal system in L2(−1, 1) of eigenfunctions is the same as (A0, D(A0)), that is {ωp}p∈N. By the strong continuity of the semigroup, see Proposition 2.4, we have that ∞∑ p=1 e−λpT1〈uε0, ωp〉ωp(x)→ uε0 in L2(−1, 1) as T1 → 0. So, there exists a small time T 1 ∈ (0, 1) such that∥∥Sε ∞∑ p=1 e−λpT1〈uε0, ωp〉ωp(·)− Sεuε0(·) ∥∥ < ε 8 , ∀T1 ∈ (0, T 1]. (3.13) EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 13 Since uε is a strict solution, by Proposition 2.18 we have f(·, ·, uε(·, ·)) ∈ L2(QT ), then using also Hölder’s inequality and Parseval’s identity we deduce that ‖Rε(x, T1)‖2 = ∞∑ p=1 ∣∣∣ ∫ T1 0 e(α1−λp)(T1−t)〈f(·, t, uε(·, t)), ωp〉dt ∣∣∣2 ≤ ∞∑ p=1 (∫ T1 0 e2(α1−λp)(T1−t)dt )∫ T1 0 |〈f(·, t, uε(·, t)), ωp〉|2dt ≤ e2α1T1T1 ∫ T1 0 ∞∑ p=1 |〈f(·, t, uε(·, t)), ωp〉|2dt = S2 εT1 ∫ T1 0 ‖f(·, t, uε(·, t))‖2dt ≤ CS2 εe 2kϑT1T 2 1 ‖uε0‖2ϑ1,a , (3.14) where C = C(‖uε0‖1,a) and k are the positive constants introduced in the statement of Proposition 2.18. Then there exists T ∗1 ∈ (0, T 1] such that √ CSεe kϑT1T1‖uε0‖ϑ1,a < √ 2 36 ε, ∀T1 ∈ (0, T ∗1 ]. (3.15) Using Proposition 2.14, by (3.12)-(3.15) and keeping in mind (3.9), for every T1 ∈ (0, T ∗1 ], we have ‖u(·, T1)− Sεuε0(·)‖ ≤ ‖u(·, T1)− uε(·, T1)‖+ ‖uε(·, T1)− Sεuε0(·)‖ ≤ e(ν+‖α + 1 ‖∞)T1‖u0 − uε0‖ + ∥∥Sε ∞∑ p=1 e−λpT1〈uε0, ωp〉ωp(·)− Sεuε0(·) ∥∥+ ‖Rε(·, T1)‖ ≤ e(ν+‖α + 1 ‖∞) √ 2 36Sεeν ε+ √ 2 36 ε+ √ CSεe kϑT1T1‖uε0‖ϑ1,a < √ 2 12 ε, (3.16) where ν ≥ 0 is given in assumption (A1). Let us set σε0(x) := u(x, T1)− Sεuε0(x), (3.17) we note that by (3.16), we have ‖σε0‖ < √ 2 12 ε . (3.18) Step 2: Steering the system from Sεu ε 0 + σε0 to u∗ at T ∈ (0, T ∗], for some T ∗ > 0. In this step let us restart at time T1 from the initial state Sεu ε 0 + σε0 and our goal is to steer the system arbitrarily close to u∗. Let us consider αε(x) := { log ( u∗ε(x) Sεuε0(x) ) for x 6= ±1, 0 for x = ±1; (3.19) 14 G. FLORIDIA EJDE-2020/59 thus by (3.11) we deduce that αε ∈ L∞(−1, 1) and αε(x) ≤ 0 for a.e. x ∈ [−1, 1]. So, there exists a sequence {αεj}j∈N ⊂ C2([−1, 1]) such that αεj(x) ≤ 0 ∀x ∈ [−1, 1], αεj(±1) = 0, αεj → αε in L2(−1, 1) as j →∞, (3.20) and lim x→±1 α′εj(x) a(x) = 0, lim x→±1 α′εj(x)a′(x) = 0. (3.21) From (3.20) we deduce eαεj(x)Sεu ε 0(x)→ eαε(x)Sεu ε 0(x) = u∗ε(x) in L2(−1, 1) as j →∞; then there exists j∗ ∈ N such that ‖eαεjSεuε0 − u∗ε‖ < ε 12 , ∀j ∈ N with j ≥ j∗ . (3.22) Let us fix an arbitrary j ∈ N with j ≥ j∗, and let us choose as control the static multiplicative function α(x, t) := 1 T − T1 αεj(x) ≤ 0 ∀(x, t) ∈ Q̃T := (−1, 1)× (T1, T ), (3.23) and call uσ(x, t) the unique strong solution that solves problem (1.1) with the following changes: • time interval (T1, T ) instead of (0, T ); • multiplicative control given by (3.23); • initial condition uσ(x, T1) = Sεu ε 0(x) + σε0(x). Let us also denote by u(x, t) the unique strict solution of the problem ut − (a(x)ux)x = αεj(x) T − T1 u+ f(x, t, u) in Q̃T := (−1, 1)× (T1, T ) B. C. u(x, T1) = Sεu ε 0(x), x ∈ (−1, 1) . (3.24) For a.e. x ∈ (−1, 1), from the equation ut(·, t) = αεj(·) T − T1 u(·, t) + ((a(·)ux(·, t))x + f(·, t, u)) t ∈ (T1, T ), by the classical variation constants technique, we obtain a representation formula of the solution u(x, t) of (3.24), that computed at time T , for x ∈ (−1, 1), becomes u(x, T ) = eαεj(x)Sεu ε 0(x) + ∫ T T1 eαεj(x) (T−τ) T ( (a(x)ux)x(x, τ) + f(x, τ, u(x, τ)) ) dτ . Let us show that u(·, T ) → u∗ε in L2(−1, 1), as T → T1 +. Since αεj(x) ≤ 0 from that by the above formula, using Hölder’s inequality and (3.22), we deduce that ‖u(·, T )− u∗ε(·)‖2 ≤ 2‖eαεj(x)Sεuε0(x)− u∗ε‖2 + 2 ∫ 1 −1 (∫ T T1 eαεj(x) (T−τ) T ((a(x)ux)x(x, τ) + f(x, τ, u(x, τ)))dτ )2 dx ≤ ε2 72 + (T − T1)‖ ( a(·)ux ) x + f(·, ·, u)‖2 L2(Q̃T ) . (3.25) EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 15 In Step 3 below, as an appendix to this proof, we will prove that the norm at right-hand side of (3.25) is bounded as T → T1 +. More precisely, we will find that ‖ ( a(·)ux ) x + f(·, ·, u)‖L2(Q̃T ) ≤ K, for a.e. T ∈ (T1, T1 + 1), (3.26) where K = K(uε0, u ∗ ε, ‖uε0‖1,a) is a positive constant. Thus, from (3.25) and (3.26) there exists T2 ∈ (T1, T1 + 1) such that for every T ∈ (T1, T2) we have ‖u(·, T )− u∗ε(·)‖ < ε 6 . (3.27) Then, using Corollary 2.15, from (3.9), (3.27) and (3.18), there exists T ∗ in the interval (T1,min{T2, T1 + 1 4ν }) such that for every T ∈ (T1, T ∗] we have ‖uσ(·, T )− u∗(·)‖ ≤ ‖uσ(·, T )− u(·, T )‖+ ‖u(·, T )− u∗ε(·)‖+ ‖u∗ε − u∗‖ ≤ √ 2‖Sεuε0 + σε0 − Sεuε0‖+ ε 6 + ε 6 < ε 2 , from which it follows the approximate controllability at any T ∈ (0, T ∗], since T1 > 0 was arbitrarily small. Moreover, if T > T ∗ using the above argument we first obtain the approximate controllability at time T ∗. Then, we restart at time T ∗ close to u∗, and we stabilize the system into the neighborhood of u∗, applying the above strategy n times, for some n ∈ N, on n small time interval by measure T−T∗ n , steering the system in every interval from a suitable approximation of u∗ to u∗. Step 3: Evaluation of ‖ ( a(·)ux ) x + f(·, ·, u)‖2 L2(Q̃T ) : Proof of the inequality (3.26). Multiplying by ( a(x)ux ) x the equation in (3.24), integrating over Q̃T = (−1, 1) × (T1, T ) and applying Young’s inequality we have ‖ ( a(·)ux ) x ‖2 L2(Q̃T ) ≤ ∫ T T1 ∫ 1 −1 ut ( a(x)ux ) x dx dt− 1 T − T1 ∫ T T1 ∫ 1 −1 αεj(x)u ( a(x)ux ) x dx dt + 1 2 ∫ T T1 ∫ 1 −1 f2(x, t, u) dx dt+ 1 2 ∫ T T1 ∫ 1 −1 ∣∣(a(x)ux ) x ∣∣2 dx dt . (3.28) Thus, by (3.28) using Proposition (2.18) we obtain ‖ ( a(·)ux ) x + f(·, ·, u)‖2 L2(Q̃T ) ≤ 2 ( ‖ ( a(·)ux ) x ‖2 L2(Q̃T ) + ‖f(·, ·, u)‖2 L2(Q̃T ) ) ≤ 4 ∫ T T1 ∫ 1 −1 ut ( a(x)ux ) x dx dt− 4 T − T1 ∫ T T1 ∫ 1 −1 αεj(x)u ( a(x)ux ) x dx dt + 4C2S2ϑ ε e2kϑ(T−T1)(T − T1)‖uε0‖2ϑ1,a, (3.29) where C = C(‖uε0‖1,a) and k are the positive constants given by Proposition 2.18. Let us estimate the first two terms of the right-hand side of (3.28). Without loss of generality, let us consider the (WDeg) problem with β0γ0 6= 0. Integrating by 16 G. FLORIDIA EJDE-2020/59 parts and using the sign condition β0β1 ≤ 0 and γ0γ1 ≥ 0 we have∫ T T1 ∫ 1 −1 ut ( a(x)ux ) x dx dt = ∫ T T1 [ut ( a(x)ux ) ]1−1dt− 1 2 ∫ T T1 ∫ 1 −1 a(x) ( u2x ) t dx dt ≤ Sε 2 γ1 γ0 a2(1)(uε0x(1))2 − Sε 2 β1 β0 a2(−1)(uε0x(−1))2 + S2 ε 2 ∫ 1 −1 a(x)(uε0x)2 dx dt = c1(Sε, u ε 0) + c2(Sε)|uε0|21,a, (3.30) where c1(Sε, u ε 0) ≥ 0 and c2(Sε) > 0 are two constants. Let us note that in the (SDeg) case or in the (WDeg) problem with β0γ0 = 0, we obtain a similar estimate, but in the third line of (3.30) at least one of the two boundary contributions is zero. Furthermore, using (3.21) and Proposition 2.14 we obtain∫ T T1 ∫ 1 −1 αεj(x)u ( a(x)ux ) x dx dt = − ∫ T T1 ∫ 1 −1 αεj(x)a(x)u2x dx dt− 1 2 ∫ T T1 ∫ 1 −1 α′εj(x)a(x) ( u2 ) x dx dt ≥ −1 2 ∫ T T1 [ α′εj(x)a(x)u2 ]1 −1 dt + 1 2 ∫ T T1 ∫ 1 −1 ( α′′εj(x)a(x) + α′εj(x)a′(x) ) u2 dx dt ≥ −1 2 sup x∈[−1,1] ∣∣α′′εj(x)a(x) + α′εj(x)a′(x) ∣∣ ∫ T T1 ∫ 1 −1 u2 dx dt ≥ −(T − T1)c(α′εj , α ′′ εj)e ν(T−T1)S2 ε‖uε0‖2 . (3.31) Finally, using (3.29)-(3.31), we prove (3.26), that is for almost every T ∈ (T1, T1+1) we have ‖ ( a(·)ux ) x + f(·, ·, u)‖2 L2(Q̃T ) ≤ k1(Sε, u ε 0) + k2(Sε)|uε0|21,a + k3(ν, Sε, α ′ εj , α ′′ εj)‖uε0‖2 + k4(Sε, ‖uε0‖1,a)‖uε0‖2ϑ1,a ≤ k1(Sε, u ε 0) +K2(ν, Sε, α ′ εj , α ′′ εj , ‖uε0‖1,a)‖uε0‖21,a, where k1 ≥ 0 and k2, k3, k4,K2 > 0 are constants. � 4. Energy balance models in climate science Climate depends on several variables and parameters such as temperature, hu- midity, wind intensity, the effect of greenhouse gases, and so on. It is also affected by a complex set of interactions in the atmosphere, oceans and continents, that involve physical, chemical, geological and biological processes. One of the first at- tempts to model the effects of the interaction between large ice masses and solar radiation on climate is the one done, independently, by Budyko [11] and Sellers [48]. A complete treatment of the mathematical formulation of the Budyko-Sellers model has been obtained by Diaz and collaborators in [23]–[28] and [9]; see also the EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 17 interesting recent monograph [45] on “Energy Balance Climate Models” by North and Kim, and some papers by Cannarsa and coauthors [13, 19]. The Budyko-Sellers model is an energy balance model, which studies the role played by continental and oceanic areas of ice on the evolution of the climate. The effect of solar radiation on climate can be summarized in Figure 1. Figure 1. Copyrighted by ASR We have the energy balance: Heat variation = Ra −Re +D, where Ra is the absorbed energy, Re is the emitted energy and D is the diffusion part. If we represent the Earth by a compact two-dimensional manifold without boundary M, the general formulation of the Budyko-Sellers model is as follows c(X, t)ut(X, t)−∆Mu(X, t) = Ra(X, t, u)−Re(u), (4.1) where c(X, t) is a positive function (the heat capacity of the Earth), u(X, t) is the annually (or seasonally) averaged Earth surface temperature, and ∆M is the classical Laplace-Beltrami operator. To simplify (4.1), we assume that the thermal capacity is c ≡ 1. Re(u) denotes the Earth radiation, that is, the mean emitted energy flux, that depends on the amount of greenhouse gases, clouds and water vapor in the atmosphere and may be affected by anthropo-generated changes. In the literature there are different empiric expressions of Re(u). In [48], Sellers proposes a Stefan-Boltzman type radiation law: Re(u) = ε(u)u4, where u is measured in Kelvin degrees (and thus u > 0), the positive function ε(u) = σ ( 1 −m tanh( 19u6 106 ) ) represents the emissivity, σ is the emissivity constant and m > 0 is the atmospheric opacity. In its place, in [11] Budyko considers a Newtonian linear type radiation, that is, Re(u) = A + Bu, with suitable A ∈ R, B > 0, which is a linear approximation of the above law near the actual mean 18 G. FLORIDIA EJDE-2020/59 temperature of the Earth, u = 288.15◦ K (15◦ C). Ra(X, t, u) denotes the fraction of the solar energy absorbed by the Earth and is assumed to be of the form Ra(X, t, u) = QS(X, t)β(u), in both the models. In the above relation, Q is the Solar constant, S(X, t) is the distribution of solar radiation over the Earth, in seasonal models (when the time scale is smaller) S is a positive “almost periodic” function in time (in particular, it is constant in time, S = S(X), in annually averaged models, that is, when the time scale is long enough), and β(u) is the planetary coalbedo representing the fraction absorbed according the average temperature (β(u) ∈ [0, 1]) The coalbedo function is equal to 1-albedo function. In climate science the albedo (see Figure 2) is more used and well-known than the coalbedo, and is the reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it. It may also be expressed as a percentage, and is measured on a scale from 0, for no reflecting power of a perfectly black surface, to 1, for perfect reflection of a white surface. The coalbedo is assumed to be a non-decreasing function of u, that is, over ice-free zones (like oceans) the coalbedo is greater than over ice- covered regions. Denoted with us = 263.15◦ K (−10◦ C) the critical value of the temperature at which ice becomes white (the “snow line”), given two experimental values ai and af , such that 0 < ai < af < 1. Budyko [11] proposed the following coalbedo function, discontinuous at us, β(u) = { ai, over ice-covered {X ∈M : u(X, t) < us}, af , over ice-free {X ∈M : u(X, t) > us} . Sellers[48] proposed a more regular (at most Lipschitz continuous) function of u. Indeed, Sellers represents β(u) as a continuous piecewise linear function (between ai and af ) with greatly increasing rate near u = us, such that β(u) = ai, if u(X, t) < us − η and β(u) = af , if u(X, t) > us + η, for some small η > 0. If we assume that M is the unit sphere of R3, the Laplace-Beltrami operator becomes ∆Mu = 1 sinφ { ∂ ∂φ ( sinφ ∂u ∂φ ) + 1 sinφ ∂2u ∂λ2 } , where φ is the colatitude and λ is the longitude. Figure 2. Copyrighted by ABC Columbia Thus, if we take the average of the temperature at x = cosφ (see in Figure 3, that the distribution of the temperature at the same colatitude can be considered EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 19 approximately uniform). In such a model, the sea level mean zonally averaged temperature u(x, t) on the Earth, where t still denotes time, satisfies a Cauchy- Neumann strongly degenerate problem, in the bounded domain (−1, 1), of the type ut − ( (1− x2)ux ) x = α(x, t)β(u) + f(x, t, u), x ∈ (−1, 1), lim x→±1 (1− x2)ux(x, t) = 0, t ∈ (0, T ) . Then, the uniformly parabolic equation (4.1) has been transformed into a 1-D degenerate parabolic equation. So, we have showed that our degenerate reaction- diffusion system (1.1) reduces to the 1-D Budyko-Sellers model when a(x) = 1−x2. Environmental aspects. We remark that the Budyko-Sellers model studies the ef- fect of solar radiation on climate, so it takes into consideration the influence of “greenhouse gases” on climate. These cause “global warming” which, consequently, provokes the increase in the average temperature of the Earth’s atmosphere and of oceans. This process consists of a warming of the Planet Earth by the action of greenhouse gases, compounds present in the air in a relatively low concentration (carbon dioxide, water vapour, methane, etc.). Greenhouse gases allow solar radia- tion to pass through the atmosphere while obstructing the passage towards space of a part of the infrared radiation from the Earth’s surface and from the lower atmo- sphere. The majority of climatologists believe that Earth’s climate is destined to change, because human activities are altering atmosphere’s chemical composition. In fact, the enormous anthropogenic emissions of greenhouse gases are causing an increase in the Earth’s temperature, consequently, provoking profound changes in the Planetary climate. One of the aims of this kind of research is to estimate the possibility of controlling the variation of the temperature over decades and cen- turies and it proposes to provide a study of the possibility of slowing down global warming. Related open problems. Keeping in mind the meaning of the multiplicative control α in the climate framework, since in the main control result of this paper, Theorem 1.4, the action must be realized over any latitude x in [−1, 1], it would be more realistic follow up in future papers the formulation that was already proposed by Von Neumann in 1955 (See Diaz [27] for a comprehensive presentation), that is, by using a localized control defined merely for some set of latitudes. From the Figure 3. Copyrighted by Edu-Arctic.eu 20 G. FLORIDIA EJDE-2020/59 multiplicative controllability point of view, that problem is hard but it is under research by Diaz and the author; one possible approach consists in following some ideas introduced in [33], in the case of uniformly parabolic equations. 5. Appendix: Proofs of existence and uniqueness results In the first subsection we prove Theorem 2.11, which shows that for all α ∈ L∞(QT ) piecewise static functions, there exists a unique strict solution u ∈ H(QT ) to (1.1), for all initial state u0 ∈ H1 a(−1, 1). Then in subsection 5.2 we can prove Theorem 2.12, that is by an approximation argument we obtain the existence and uniqueness of the strong solution to (1.1), for all initial state u0 ∈ L2(−1, 1). 5.1. Existence and uniqueness of the strict solution to (1.1). The aim is to prove Theorem 2.11. For this purpose, we will follow the following strategy: • in Subsection 5.1.1 we present a maximal regularity result for abstract non- homogeneous linear evolution equations in Hilbert spaces; • in Subsection 5.1.2 we introduce the notion of mild solutions and we give an existence and uniqueness result for mild solutions; • in Subsection 5.1.3 we prove the existence and uniqueness of strict solutions for static coefficient α ∈ L∞(−1, 1); • in Subsection 5.1.4, finally we prove that if u0 ∈ H1 a(−1, 1) then the mild solution is also a strict solution, for all α ∈ L∞(QT ) piecewise static func- tion. 5.1.1. A maximal regularity result for linear problems. Let us consider the following linear problem in the Hilbert space L2(−1, 1), u′(t) = Au(t) + g(t), t > 0 u(0) = u0 , (5.1) where A is the operator in (2.2), g ∈ L1(0, T ;L2(−1, 1)), u0 ∈ L2(−1, 1). First, let us recall the notion of “weak solution” introduced by Ball [4] for the linear problem (5.1) (see also [12, 1, 35, 36]). Definition 5.1. Aweak solution of (5.1) is a function u ∈ C([0, T ];L2(−1, 1)) such that for every v ∈ D(A∗) (A∗ denotes the adjoint of A) the function 〈u(t), v〉 is absolutely continuous on [0, T ] and d dt 〈u(t), v〉 = 〈u(t), A∗v〉 + 〈g(t), v〉, for almost all t ∈ [0, T ]. Then, we recall the following existence and uniqueness result obtained by Ball [4] (see also [13, 14, 35, 36]). Proposition 5.2. For every u0 ∈ L2(−1, 1) there exists a unique weak solution u of (5.1), which is given by the following representation formula u(t) = etAu0 + ∫ t 0 e(t−s)Ag(s) ds, t ∈ [0, T ]. Now, we are able to present Proposition 5.3, a maximal regularity result that holds in the Hilbert space L2(−1, 1). Before giving the statement of Proposition 5.3 we recall that by maximal regularity for (5.1) we mean that u′ and Au have the same regularity of g. EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 21 Proposition 5.3. Let T > 0 and g ∈ L2(0, T ;L2(−1, 1)). For each u0 ∈ H1 a(−1, 1), there exists a unique solution of (5.1), u ∈ H(QT ) = L2(0, T ;D(A0)) ∩H1(0, T ;L2(−1, 1)) ∩ C([0, T ];H1 a(−1, 1)) . Moreover, there exists a positive constant C0(T ) (nondecreasing in T ), such that ‖u‖H(QT ) ≤ C0(T )[‖u0‖1,a + ‖g‖L2(QT )]. Proof. It is a direct consequence of [8, Theorem 3.1, Section 3.6.3 ] (see also [21, 35]), keeping in mind the following three crucial issues related to the abstract setting in [8, Section 3.6.3]: • g ∈ L2(0, T ;X), where X is the Hilbert space L2(−1, 1); • A is the infinitesimal generator of an analytic semigroup (see [20]); • u0 ∈ H1 a(−1, 1), where H1 a(−1, 1) is an interpolation space between the domain D(A0) and L2(−1, 1). � 5.1.2. Existence and uniqueness of the mild solution to (1.1). Before introducing the notion of mild solutions, we consider the following abstract representation of the semilinear problem (1.1) in the Hilbert space L2(−1, 1), u′(t) = A0 u(t) + ψ(t, u(t)) , t > 0 u(0) = u0 ∈ L2(−1, 1) , (5.2) where A0 is the operator defined in (2.1) and, for every u ∈ B(QT ), ψ(t, u) := ψ(x, t, u(x, t)) = α(x, t)u(x, t) + f(x, t, u(x, t)), ∀(x, t) ∈ QT , (5.3) with α ∈ L∞(QT ), piecewise static, given in (1.1). We note that, since from (1.3), for a.e. (x, t) ∈ QT , ∀u, v ∈ R, it follows that |f(x, t, u)− f(x, t, v)| ≤ ν ( 1 + |u|ϑ−1 + |v|ϑ−1 ) |u− v|, we deduce that |ψ(t, u)− ψ(t, v)| ≤ |α(x, t)u− α(x, t)v|+ |f(x, t, u)− f(x, t, v)| ≤ ‖α‖∞|u− v|+ ν ( 1 + |u|ϑ−1 + |v|ϑ−1 ) |u− v|. Therefore, |ψ(t, u)− ψ(t, v)| ≤ L|u− v|, for a.e. (x, t) ∈ QT , ∀u, v ∈ L2(−1, 1), (5.4) where L = L(u, v) = ‖α‖∞ + ν ( 1 + |u|ϑ−1 + |v|ϑ−1 ) which does not depend on t. Definition 5.4. Let u0 ∈ L2(−1, 1). We say that u ∈ C([0, T ];L2(−1, 1)) is a mild solution of (1.1), if u is a solution of the integral equation u(t) = etA0u0 + ∫ t 0 eA0(t−s)ψ(s, u(s)) ds, t ∈ [0, T ]. The existence and the uniqueness of the mild solution of (1.1) follows from Proposition 5.5 below, that is a consequence of a result by Li and Yong [44, Propo- sition 5.3 in Chapt. 2]. For the next proposition the following assumptions are introduced: 22 G. FLORIDIA EJDE-2020/59 (A3) for each ū ∈ L2(−1, 1), ψ(·, ū) : [0, T ] → L2(−1, 1) is strongly measur- able, that is there exists a sequence of simple functions (piecewise static functions) ψk(·, ū) : [0, T ]→ L2(−1, 1) such that lim k→∞ |ψk(t, ū)− ψ(t, ū)| = 0, a.e. t ∈ [0, T ]; (A4) there exists a function L(t) ∈ L1(0, T ) such that |ψ(t, u)− ψ(t, v)| ≤ L(t)|u− v|, a.e. t ∈ [0, T ], ∀u, v ∈ L2(−1, 1), |ψ(t, 0)| ≤ L(t), a.e. t ∈ [0, T ]. Proposition 5.5. Under assumptions (A3) and (A4), There exists a unique mild solution to (5.2). 5.1.3. Existence and uniqueness of strict solutions for static coefficient α ∈ L∞(−1, 1). In this subsection we prove Theorem 2.11 by showing that if the initial state belongs to H1 a(−1, 1), then the mild solution is also a strict solution. Lemma 5.6. For every M > 0, there exists TM > 0 such that for all α ∈ L∞(−1, 1), and all u0 ∈ H1 a(−1, 1) with ‖u0‖1,a ≤ M there is a unique strict solution u ∈ H(QTM ) to (1.1). Proof. Let us fix M > 0, u0 ∈ H1 a(−1, 1) such that ‖u0‖1,a ≤ M . Let 0 < T ≤ 1, we define HM (QT ) := {u ∈ H(QT ) : ‖u‖H(QT ) ≤ 2C0(1)M}, where C0(1) is the constant C0(T ) (nondecreasing in T ) defined in Proposition 5.3 and valued in T = 1. Then, let us consider the map Φ : HM (QT ) → HM (QT ), defined by Φ(u)(t) := etA0u0 + ∫ t 0 e(t−s)A0 (αu(s) + f(s, u(s))) ds ∀u ∈ H(QT ). Step 1. We prove that the map Φ is well defined for some T . Fix u ∈ HM (QT ), and consider y(t) := Φ(u)(t), then, keeping in mind Proposition 5.2, we can see the function y as the solution of the problem y′(t) = A0y(t) + (αu(t) + f(t, u(t))) , t ∈ [0, T ] y(0) = u0 ∈ H1 a(−1, 1) (5.5) By Proposition 2.18 we deduce that f(·, ·, u) ∈ L2(QT ), thus g(t) := ψ(t, u(t)) ∈ L2(0, T ;L2(−1, 1)). Then, applying Proposition 5.3 there exists a unique solution y ∈ H(QT ) of (5.5) such that ‖y‖H(QT ) ≤ C0(T ) ( ‖u0‖1,a + ‖ψ(·, u(·))‖L2(QT ) ) . Thus, keeping in mind Proposition 5.3, we have C0(T ) ≤ C0(1) since T ≤ 1. Applying Lemma 2.7 and Proposition 2.18 there exists T0(M) ∈ (0, 1) such that ‖Φ(u)‖H(QT ) = ‖y‖H(QT ) ≤ C0(1) ( ‖α‖∞‖u‖L2(QT ) + ‖f(·, ·, u)‖L2(QT ) + ‖u0‖1,a ) ≤ 2C0(1)M, ∀T ∈ [0, T0(M)]. Then Φu ∈ HM (QT ) for all T ∈ [0, T0(M)]. EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 23 Step 2.; We prove that there exists TM such that the map Φ is a contraction. Let T ∈ (0, T0(M)]. Fix u, v ∈ HM (QT ) and set w := Φ(u)− Φ(v) = ∫ t 0 e(t−s)A0 [α (u(s)− v(s)) + (f(s, u(s))− f(s, v(s)))] ds . Then, keeping in mind Proposition 5.2, w is solution of the problem wt − (awx)x = ψ(t, u)− ψ(t, v) in QT B. C. w(x, 0) = 0 . (5.6) By Lemma 2.18 f(·, u) ∈ L2(QT ), and applying Proposition 5.3 we deduce that there exists a unique solution w ∈ H(QT ) of (5.6), and ‖Φ(u)− Φ(v)‖H(QT ) = ‖w‖H(QT ) ≤ C0(1)‖ψ(·, u)− ψ(·, v)‖L2(QT ). (5.7) From (5.7), using (5.4), Lemma 2.7 and Proposition 2.18 (for similar estimates see [35, Lemma B.1]), there exists TM ∈ (0, T0(M)) such that Φ is a contraction map. Therefore, Φ has a unique fix point in HM (QTM ), from which the conclusion follows. � Remark 5.7. Using a classical “semigroup” result (see [46]), applying Lemma 5.6 and the “a priori estimate” contained in Lemma 2.17, we directly obtain the existence of the global strict solution to (1.1). That is, following the proof of Lemma 5.6 the unique mild solution, given by Proposition 5.5, is also a strict solution. Thus, we obtain the proof of Theorem 2.11 in the case of a static reaction coefficient α ∈ L∞(−1, 1). 5.1.4. Regularity of the mild solution to (1.1) with initial data in H1 a(−1, 1). Now, we can give the complete proof of Theorem 2.11 in the general case when α ∈ L∞(QT ) is a piecewise static function. Proof of Theorem 2.11. Let us consider (1.1) under assumptions (A1) and (A2). Let us assume that u0 ∈ H1 a(−1, 1) and α ∈ L∞(QT ) is a piecewise static function (or a simple function with respect to the variable t), in the sense of Definition 1.1, that is, there exist m ∈ N, αk(x) ∈ L∞(−1, 1) and tk ∈ [0, T ], tk−1 < tk, k = 1, . . . ,m with t0 = 0 and tm = T , such that α(x, t) = α1(x)χ[t0,t1](t) + m∑ k=2 αk(x)χ(tk−1,tk](t), where χ[t0,t1] and χ(tk−1,tk] are the indicator function of [t0, t1] and (tk−1, tk], re- spectively. Let u ∈ C([0, T ];L2(−1, 1)) the unique mild solution of (1.1) with initial state u0 ∈ H1 a(−1, 1), given by Proposition 5.5, u(t) := etA0u0 + ∫ t 0 e(t−s)A0 (α(s)u(s) + f(s, u(s))) ds , ∀t ∈ [0, T ]. Then u, for k = 1, · · · ,m, is the solution of the following m problems: U ′(t) = A0 U(t) + αkU(t) + f(t, U(t)) , t ∈ [tk−1, tk] U(tk) = u(tk) k = 1, . . . ,m . (5.8) Since u0 ∈ H1 a(−1, 1) and αk ∈ L∞(−1, 1) (k ∈ {1, · · · ,m}) is static on [tk−1, tk], then applying m times Remark 5.7 we obtain that the unique mild solution u is also 24 G. FLORIDIA EJDE-2020/59 strict on [tk−1, tk], then the “new” initial condition u(tk) will belong to H1 a(−1, 1). Thus, by iteration from [0, t1] to [tm−1, tm] we can complete the proof and we obtain that the mild solution u ∈ H(QT ) and it is also a strict solution on [0, T ]. � Remark 5.8. Keeping in mind Proposition 5.5 it follows that we can extend The- orem 2.11 to the general case α ∈ L∞(QT ). Namely, in that case α is strongly measurable, in the sense of the condition (A4), moreover we can generalize the proof of Theorem 2.11 from α piecewise static function to α strongly measurable. Remark 5.9. To prove Theorem 2.11 one can also follow the approach developed by Kato [40] and Evans [31] (see also Pazy’s book [46]). This approach considers in (5.2) directly as operator A(t) := A0 + α(t)I, in which the dependence on t is discontinuous, instead of the simple operator A0 that is constant in t. 5.2. Existence and uniqueness of the strong solution to (1.1). In this section we recall the proof of Theorem 2.16, given in [35] for the (SDeg) case, and in [36] for the (WDeg). Proof of Theorem 2.16. Since u0 ∈ L2(−1, 1), there exists {u0k}k∈N ⊆ H1 a(−1, 1) such that limk→∞ u0k = u0 in L2(−1, 1). For every k ∈ N, we consider the problem ukt − (a(x)ukx)x = α(x, t)uk + f(x, t, uk) a.e. in QT B. C. uk(x, 0) = u0k(x) x ∈ (−1, 1) . (5.9) For every k ∈ N, by Theorem 2.11 there exists a unique uk ∈ H(QT ) strict solution to (5.9). Then, we consider the sequence {uk}k∈N ⊆ H(QT ) and by direct applica- tion of Proposition 2.14 we prove that {uk}k∈N is a Cauchy sequence in the Banach space B(QT ). Then, there exists u ∈ B(QT ) such that, as k →∞, uk → u in B(QT ) and u(·, 0) = limk→∞ uk(·, 0) = u0 in the L2 sense. So, u ∈ B(QT ) is a strong so- lution. The uniqueness of the strong solution to (1.1) is a direct consequence of Proposition 2.14. � Acknowledgments. The author is indebted to Prof. Ildefonso Diaz both for the suggestions about mathematical point of view for “Energy Balance Climate Mod- els”, and for the criticism and the useful comments which have made this paper easier to read and understand. 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Vancostenoble; Global Non-negative Approximate Controllability of Parabolic Equations with Singular Potentials, Springer INdAM series, 32, 2019, doi:10.1007/978-3-030-17949-6, Trends in Control Theory and Partial Differential Equations, by F. Alabau-Boussouira, F. Ancona, A. Porretta, C. Sinestrari. EJDE-2020/59 NONNEGATIVE CONTROLLABILITY 27 Giuseppe Floridia Università Mediterranea di Reggio Calabria, Dipartimento Patrimonio, Architettura, Urbanistica, Via dell’Università 25, Reggio Calabria, 89124, Italy Email address: floridia.giuseppe@icloud.com 1. Introduction 1.1. Problem formulation 1.2. Main results 1.3. State of the art in multiplicative controllability and degenerate parabolic equations 2. Well-posedness 2.1. Weighted Sobolev spaces 2.2. Existence and uniqueness of solutions of semilinear degenerate problems 3. Proof of main results 3.1. Nonnegative solutions 3.2. Nonnegative controllability 4. Energy balance models in climate science 5. Appendix: Proofs of existence and uniqueness results 5.1. Existence and uniqueness of the strict solution to (1.1)to to 5.2. Existence and uniqueness of the strong solution to (1.1)to to Acknowledgments References