Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 42, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS FOR HÉNON TYPE PARABOLIC EQUATIONS WITH EXPONENTIAL NONLINEARITY CAIHONG CHANG, ZHENGCE ZHANG Abstract. This article concerns the blow up behavior for the Hénon type parabolic equation with exponential nonlinearity, ut = ∆u+ |x|σeu in BR × R+, where σ ≥ 0 and BR = {x ∈ RN : |x| < R}. We consider all cases in which blowup of solutions occurs, i.e. N ≥ 10 + 4σ. Grow up rates are established by a certain matching of different asymptotic behaviors in the inner region (near the singularity) and the outer region (close to the boundary). For the cases N > 10 + 4σ and N = 10 + 4σ, the asymptotic expansions of stationary solutions have different forms, so two cases are discussed separately. Moreover, different inner region widths in two cases are also obtained. 1. Introduction 1.1. Background. In this article, we consider the parabolic problem ut = ∆u+ λ|x|σeu in BR′ × R+, u(x, t) = 0 on SR′ × R+, u(x, 0) = u0(x) in BR′ , (1.1) where σ ≥ 0, λ > 0, BR′ = {x ∈ RN : |x| < R′} is the unit ball with the boundary SR′ = {x ∈ RN : |x| = R′}. Problem (1.1) is known in stellar structure and combustion theory [18, 30]. The geometric background of (1.1) is presented in [24, 53]. Let M be a compact manifold and pi (i = 1, 2, . . . , k) be the puncture points. If there exists a nonsingular conformal map φi from Ui (a neighborhood of pi, and Ui ∩Uj = ∅, i 6= j) to B (the unit ball) such that φi(pi) = 0 and ds2 = ρ(φi)|φi|2βi |dφi|2, then for any continuous function ρ, the point pi is a conical singularity of order βi of the metric ds2. In particular, a singular Riemannian metric on U∗i = Ui\{pi} is defined as ds2 = |x|2βids2 0, βi > −1, x ∈ B\{0}, where ds0 is the Euclidean metric. We extended to M0 = M\{p1, p2, . . . , pk} and denote it by a conical metric. Researchers also look at the asymptotic behaviors at 2020 Mathematics Subject Classification. 35A01, 35B40, 35B44, 35K20. Key words and phrases. Matched expansion; weighted term; stabilization; grow up rate; degeneracy. ©2022. This work is licensed under a CC BY 4.0 license. Submitted May 3, 2021. Published June 28, 2022. 1 2 C. CHANG, Z. ZHANG EJDE-2022/42 the singular point pi of solutions to the elliptic equation with the conical metric −∆gu = up, x ∈M0, (1.2) where N N−2 < p ≤ N+2 N−2 . At each singular point pi, the conical metric gives the Laplace-Beltrami operator ∆g in the form of ∆g = 1 |x|Nβi N∑ j=1 ∂ ∂xj ( |x|(N−2)βi ∂ ∂xj ) , x ∈ Ui\{pi}. Hence, (1.2) in Ui\{pi} can be written as the weighted equation − div(|x|γ∇u) = |x|σup, x ∈ B\{0}, (1.3) where γ = (N − 2)βi, σ = Nβi, and σ − γ = 2βi > −2. Since the function eu can be regarded as the limit of up when p → ∞, thus, (1.1) is a special case of (1.3) when γ = 0 and p→∞. In this article, we discuss the impact of the singularity at x = pi (i.e. x = 0) on the asymptotic behaviors of the solutions. The weight |x|σ is essential. To describe the dynamics of globular cluster of stars, Matukuma [37] in 1930 first proposed the model with weighted term: ∆u = 1 1+|x|2u p. Based on numerical simulations, Hénon [25] in 1973 considered the equa- tion −∆u = |x|σup, σ > 0, p > 1. (1.4) For the works on the existence and qualitative properties of solutions to (1.4), we refer to [4, 3, 7, 21, 35, 39, 42, 45, 46, 47]. In 1982, Ni [39] first realized that the weight |x|σ impacts the global homogeneity of (1.4) and widths the critical exponent between existence and nonexistence. Moreover, he proved that (1.4) admits at least one radial solution for p ∈ ( 1, N+2+2α N+2 ) . Recently, Barboza et al. [4] studied the Hénon-type Dirichlet problem, and proved existence of at least one radial solution using variational methods. Because the weight rσ is increasing with respect to r, the classical moving plane method cannot be applied to (1.4), nonradial solutions appear naturally. In [47], Smets et al. proved that ground state is nonradial for p ∈( 1, N+2 N−2 ) and sufficiently large σ. The property that |x|σ prohibits concentration phenomena at zero is applied in the case p = N+2 N−2 for sufficiently large σ in [45], and it is used to obtain multiplicity results in [3]. The problems with weighted exponential source are also studied in [9, 41, 12]. 1.2. Known results. For the stationary problem of (1.1), −∆U = λ|x|σeU in BR′ , U(x) = 0 on SR′ , (1.5) there exists a critical value of λ in the form of λ∗ = sup {λ > 0 : (1.5) admits at least one solution} . Thus, (1) If λ > λ∗, (1.5) admits no solution, that is, any solution of (1.1) blows up in the finite time. (2) If λ < λ∗, (1.5) admits a bounded classical solution, that is, the solution of (1.1) converges to stationary solution for sufficiently small initial data u0. EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 3 (3) If λ = λ∗, (1.5) admits the extremal solution U∗(x) = U(x, λ∗). The behavior of U∗(x) depends on the dimension N . When N < 10 + 4σ, U∗ is bounded, corresponding to the case of a closed spectrum. When N ≥ 10 + 4σ, (1.5) admits no classical stationary solution and ‖U(x, λ)‖∞ →∞ as λ→ λ∗, corresponding to the case of an open spectrum. In this paper, we focus on the case λ = λ∗. We study the relation between the asymptotic behavior of solutions and dimension N . The similar relation also holds for the equations with nonlinearity λeu and λ(1 + s)p with p > 1 in [18, 30, 33, 40, 11, 6] and λf(x) (1−u)2 in [19, 22]. During the past several decades, a lot of works regarding the existence and the asymptotic behavior of blowup solutions have emerged in [1, 2, 10, 57, 51, 44], see also [36, 43, 54, 55, 56, 48, 49, 50] for gradient blowup studies. Results include blowup criteria, blowup locations, blowup rates, and blowup profiles. Blowup rates are usually determined by the self-similar rates, which are related to the scaling invariance of the equation. For the classical semilinear heat equation ut = ∆u+ up in BR × R+, u = 0 on ∂BR × R+, u(x, 0) = u0(x) in BR, (1.6) the blowup rate of solutions is (T − t)− 1 p−1 for 1 < p < N+2 N−2 . This result shows that the self-similar rate is not the only rate, there are other rates which are usually refereed to as Type II rate, a vast literatures exist [14, 20, 26, 27, 28, 29]. If the large time behavior of solutions is not self-similar, the blowup profiles could be obtained by applying the matched asymptotic method, see [5, 13, 14, 15, 16, 17, 23, 32, 38, 50, 52]. For (1.6), the cases p = N+2 N−2 with N = 3, 4, 5 and p ≥ pu = N−2 √ N−1 N−4−2 √ N−1 with N > 10 were studied in [16] and [13], respectively. Galaktionov et al. [13] also considered the semilinear Frank-Kamenetskii equation ut = ∆u+ eu in BR × R+, u = 0 on ∂BR × R+, u(x, 0) = u0(x) in BR. (1.7) It was proved that for N > 10 and u0 belows the singular stationary solution Us(x), then ‖u(x, t)‖∞ = α0t+O(1), t→∞, where α0 = α0(N) > 0. Galaktionov and King [17] studied the critical case N = 10, and showed that ‖u(0, t)‖∞ = α0t+O(log t), t→∞, where α0 is given by the first eigenvalue of an associated linear differential operator. Matched asymptotic expansions can also be applied to study other PDE models, see [50, 34, 8, 31, 32, 52]. 1.3. Main results. Motivated by [13, 17], we consider the blowup rates of solutions of (1.1) in the critical case λ = λ∗. By rescaling x 7−→ (λ∗) 1 2+σ x, (1.1) with λ = λ∗ can be written as ut = ∆u+ |x|σeu in BR × R+, u(x, t) = 0 on SR × R+, u(x, 0) = u0(x) in BR. (1.8) 4 C. CHANG, Z. ZHANG EJDE-2022/42 It admits the explicit singular stationary solution Us(x) = log (2 + σ)(N − 2) |x|2+σ , N > 2. (1.9) By the boundary condition, we have R = [(2 + σ)(N − 2)] 1 2+σ . The initial data u0 ∈ L1(BR) satisfies u0(x) ≤ Us(x), u0(x) 6≡ Us(x). (1.10) Our main results read as follows. Theorem 1.1. Let N > 10+4σ and condition (1.10) hold. Then the global solution to (1.8) satisfies ‖u(·, t)‖L∞(BR) = (2 + σ)λ1 |γ+| t+O(1), t→∞, (1.11) where γ+ = 1 2 [ 2−N + √ (N − 2)(N − 10− 4σ) ] < 0, and λ1 is defined in Lemma 2.5 as the first eigenvalue of an associated linearized problem. Theorem 1.2. Let N = 10+4σ and condition (1.10) hold. Then the global solution to (1.8) satisfies u(0, t) = π2 1t 2(4 + 2σ) 4 2+σ +O(log t), t→∞, (1.12) where π1 is the first zero of the zeroth-order Bessel’s function, i.e. J0(π1) = 0. Based on Theorems 1.1 and 1.2, we find that the asymptotic behavior of so- lutions depends on N and σ. When N > 10 + 4σ, the lower order term of as- ymptotic expansion of stationary solution is in the form of exponential function times power function, i.e. e µγ+ 2+σ rγ+ (see (2.9)). When N = 10 + 4σ, it is in the form of exponential function times power function times logarithmic function, i.e. e−2µr−4−2σ log(re µ 2+σ ) (see (2.10)). In the matching process, logarithmical term makes it difficult to estimate the upper bound of solution. Following the general strategy of [17, 20, 23], we find the term which matches the logarithmical term. In the outer region (away from r = 0), the term rσ is a function with upper and lower bounds and strictly greater than 0, so it is evaluated as a constant. But this is not applied in the inner region (near r = 0), the degeneration of weight rσ will lead some difficulties as follows. Firstly, when the maximum of the solution is attained, the term rσeu in (1.1) would be removed due to u(0, t) = supr u(r, t), which makes it impossible to apply (1.1) to estimate. To solve this problem, we adopt the idea of limit (consider the case r → 0), and use the inner region width to characterize r, i.e. r ≤ Ce− u(0,t) 2+σ (r → 0 as t→∞). However, when σ = 0, (1.1) can be used directly to obtain the estimate due to the existence of the term eu. Secondly, the weight |x|σ generates complex calculations in the asymptotic ex- pansion of stationary solution (see Subsection 3.1). When N = 10+4σ, the asymp- totic expansions in the inner and outer regions do not match directly, see Remark EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 5 3.2. To solve this problem, we apply the method in [15] to obtain a more accurate asymptotic expansion (exact coefficients and lower order terms) in the inner region. An ODE of order (2+σ) is obtained during the calculation (see (3.3)). To solve this ODE, we assume the solution has a perturbation term Ω (i.e. (3.4)). We apply Tay- lor expansion to extract the terms with only Ω′′, Ω′ and Ω, and then solve the ODE composed of these three terms to obtain the accurate expressions of Ω (i.e. (3.8), (3.9)). Later on, we put the expressions of Ω into the remaining term G(ξ,N, σ) to verify that G(ξ,N, σ)→ 0 as ξ →∞. However, when σ = 0, Galaktionov and King [17] obtained a quadratical ODE, which can be solved directly without applying Taylor expansion. This article is organized as follows. In Section 2, we consider the case N > 10+4σ and prove Theorem 1.1. In Section 3, we study the case N = 10 + 4σ and prove Theorem 1.2. 2. The case N > 10 + 4σ 2.1. Preliminary estimates. By (1.10), the maximum principle implies that u(x, t) ≤ Us(x), u(x, t) 6≡ Us(x). Set w(x, t) = Us(x)− u(x, t). (2.1) Clearly, w(x, t)→ 0 as t→∞. We find that w satisfies wt = ∆w + ν |x|2 (1− e−w) in BR × R+, w(x, t) = 0 on SR × R+, w(x, 0) = w0(x) in BR, (2.2) where ν = (2 + σ)(N − 2) and w0(x) ∈ L1(BR). Applying a standard regularity theory, we deduce that w(x, t) ∈ C∞ ( BR \ {0} ×R+ ) , it makes sense that w0(x) ∈ L2(BR). Next, we give useful lemmas which are similar to the ones in [13], the proofs are valid to ours with few modifications, so we omit them. Lemma 2.1. Let N ≥ 10 + 4σ. Then as t→∞, we have ‖w(·, t)‖2 ≤ ce−mt, where m = m(N) > 0. By Lemma 2.1, the following lemma provides a linear lower bound of u, which plays a key role in the inner analysis. Lemma 2.2. Let N ≥ 10 + 4σ and condition (1.10) hold. Then ‖u(·, t)‖∞ ≥ 2m N t(1 + o(1)), (2.3) where m = m(N) > 0. 2.2. Asymptotic behavior in the inner region. 6 C. CHANG, Z. ZHANG EJDE-2022/42 2.2.1. Properties of radial stationary solutions for N ≥ 10 + 4σ. We consider the radially symmetric stationary equation ∆U + rσeU = 0, U = U(r), r > 0. (2.4) Set S(U) = ∆U + rσeU . Let U0(r) be the solution of (2.4) under the assumptions U0(0) = 0, U ′0(0) = 0. Clearly, U0(r) < 0 and U ′0(r) < 0 for all r > 0. In fact, it follows from (2.4) that (rN−1U ′0)′ = −rN−1+σeU0 < 0, that is, the term rN−1U ′0 is decreasing with respect to r. Recalling that U ′0(0) = 0, we have U ′0(r) < 0. By U0(0) = 0, we obtain U0(r) < 0. When N ≥ 10 + 4σ, for r > 0, we have U0(r) < Us(r) = log (2 + σ)(N − 2) |x|2+σ . (2.5) Moreover, if N > 10 + 4σ, then as r →∞, U0(r) = Us(r)− b0rγ+ ( 1 + o(1) ) , (2.6) where b0 = b0(σ,N) > 0, γ+ = 1 2 [ 2−N + √ (N − 2)(N − 10− 4σ) ] < 0. If N = 10 + 4σ, then as r →∞, U0(r) = Us(r)− b0r−4−2σ log r ( 1 + o(1) ) (2.7) where b0 = b0(σ,N) > 0. Notice that the asymptotic expansion of U0(r) includes the logarithmic term. For any given µ ∈ R, let Uµ(r) be the solution of (2.4) under the assumptions Uµ(0) = µ, U ′µ(0) = 0. By the scaling invariance of (2.4), we deduce Uµ(r) = µ+ U0 ( re µ 2+σ ) . (2.8) The maximum principle implies that Uµ(r) < Us(r). If N > 10 + 4σ, there exists a sufficiently small δ such that, for r ≥ δ, Uµ(r) = Us(r)− b0e µγ+ 2+σ rγ+ ( 1 + o(1) ) , µ→∞. (2.9) If N = 10 + 4σ, we have for r ≥ δ, Uµ(r) = Us(r)− b0e−2µr−4−2σ log ( re µ 2+σ )( 1 + o(1) ) , µ→∞. (2.10) Summing up all the above cases, we find that as µ→∞, Uµ(r)→ Us(r) unformly on [δ,∞). Moreover, the solution Uµ(r) is strictly monotone increasing with respect to µ for all r ≥ 0. In Subsection 3.1, we will give more accurate asymptotic expansions of stationary solutions. In particular, we obtain the lower order term which is in the form of power functions, and the precise expression of coefficient b0 in (2.10). EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 7 2.2.2. Inner analysis. We consider the asymptotic behavior of solutions in the in- ner region, which is a small region near x = 0 for sufficiently large t. Based on symmetrization and comparison argument, we suppose that u = u(r, t) ≥ 0 is sym- metric and decreasing with respect to r for all t ≥ 0. It follows from Lemma 2.2 that α(t) = sup r u(r, t) = u(0, t)→∞, t→∞. (2.11) By intersection comparison with stationary, we find that α(t) is strictly monotonous increasing with respect to t, that is, α′(t) > 0, t→∞. (2.12) Theorem 2.3. Let N ≥ 10 + 4σ. Then as t→∞, u(r, t) = Uα(t)(r) ( 1 + o(1) ) (2.13) uniformly on compact subsets {ξ = re α(t) 2+σ ≤ C} with C > 0. To prove Theorem 2.3, we introduce the rescaled function θ, which satisfies u(r, t) = α(t) + θ(ξ, τ), ξ = re α(t) 2+σ . (2.14) It follows from (2.11) that θ(0, τ) ≡ 0, θ ≤ 0. (2.15) We set the new time variable τ in the form of τ = ∫ t 0 e 2α(s) 2+σ ds→∞, t→∞. (2.16) Substituting (2.14) into (1.1), we obtain that θ(ξ, τ) satisfies θτ = S(θ) + g(τ) [ 1 2 + σ θξξ + 1 ] , (2.17) where the operator S is defined in (2.4) and g(τ) = −α′(t)e− 2α(t) 2+σ = [2 + σ 2 e− 2α(t) 2+σ ]′ < 0. (2.18) Equation (2.17) can be viewed as the time-dependent perturbation of (1.1). It follows from (2.18) and (2.16) that∫ ∞ g(τ)dτ = ∫ ∞ g(τ) dτ dt dt = ∫ ∞ −α′(t)e− 2α(t) 2+σ e 2α(t) 2+σ dt = − ∫ ∞ α′(t)dt = −∞, i.e. g /∈ L1(R+), that is, the perturbation g(τ) is not integrable in time. However, the following lemma ensures that the perturbation vanishes as τ →∞. Lemma 2.4. It holds limτ→∞ g(τ) = 0. Proof. We claim that g(τ) is uniformly bounded on compact subsets of ξ. In fact, by (2.11) and (2.14), α′(t) = ut(0, t) ≤ lim r→0 rσeu ≤ lim r→0 rσeα(t) = lim r→0 ξσe− σα(t) 2+σ eα(t) ≤ Cσe 2α(t) 2+σ on any compact subsets of ξ. It follows from (2.18) that |g(τ)| ≤ Cσ. Next, we prove that g(τ)→ 0 as τ →∞ by contradiction. Since g(τ) is uniformly bounded, we may assume that there exists a sequence {τk}k∈N → ∞ such that 8 C. CHANG, Z. ZHANG EJDE-2022/42 g(τk)→ −γ0 < 0. By the standard regularity, we deduce that θ(·, τk + s)→ h(·, s), where h satisfies hs = S(h)− γ0 [ 1 2 + σ hξξ + 1 ] , s ≥ 0, (2.19) and h(0, s) ≡ 0, h(ξ, s) ≤ Us(ξ). Let V0 be the solution of the stationary problem of (2.19), that is, S(V0)− γ0 [ 1 2 + σ V0ξξ + 1 ] = 0, V0(0) = 0. The maximum principle implies that V0 ≤ Us. The function V0 comes from the self-similar solution u∗ of (1.1) with finite time T in the form of u∗(x, t) = −2 + σ 2 log [ 2γ0 2 + σ (T − t) ] + V0(η), η = x√ 2γ0 2+σ (T − t) . (2.20) We shall obtain that V0 must intersect Us by contradiction. Assume that V0 < Us. It follows from (1.10) and the maximum principle that u∗ ≤ Us and u∗ 6≡ Us. Recalling (2.20), V0 < Us and the fact − 2+σ 2 log[ 2γ0 2+σ (T − t)] → +∞ as t → T , we deduce that u∗ ≤ Us is not valid. Thus, g(τ) vanishes at infinity. � Proof of Theorem 2.3. Fix a sequence {τk}k∈N → ∞. Let θ = θ(·, τk + s). Apply- ing the standard interior regularity, we obtain that θ, θξ, θξξ, θτ , θτξ are uniformly bounded and θ(·, τk + s) → f(·, s) uniformly on any compact sets of ξ. Then f satisfies the limit equation of (2.17), i.e. fs = S(f) in R+ × R+ , (2.21) f(0, s) = 0, f ≤ 0. (2.22) Since coefficients in (2.21) are analytic, by the standard regularity theory, we know that f is a C∞ function and analytic in ξ. Notice that (2.13) is equivalent to f(ξ, s) ≡ U0(ξ). We proceed by contradiction. Assume that f(ξ, s) 6≡ U0(ξ). By (2.22) and U0(0) = 0, we have that f(ξ, s) intersects U0(ξ) infinitely many times for all s ≥ 0. On the other hand, based on the Sturmian argument, the number of intersections between θ(ξ, τ) and U0(ξ) cannot increase with respect to time, and it is finite initially since the solutions are analytic. This leads to a contradiction. � The following lemma shows that the stabilization in (2.13) is from above, see details in [13, Lemma 3.2]. Lemma 2.5. Let N ≥ 10 + 4σ. Then θ(ξ, t) ≥ U0(ξ), t→∞. (2.23) 2.3. Asymptotic behavior in the outer region. We consider the asymptotic behavior of solutions in the outer region, which is the region away from x = 0 for t sufficiently large. EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 9 2.3.1. Linearized analysis. (2.2) can be written as wt = −Aw − F (w), (2.24) where A is the linear operator in the form of Aw = −∆w − ν |x|2 w (2.25) with ν = (2 + σ)(N − 2), and F is the nonlinear operator in the form of F (w) = ν |x|2 (e−w − 1 + w) ≥ 0 for w ≥ 0. (2.26) The corresponding radially homogeneous problem of (2.24) is given by Aψ = 0 in BR. If N > 10 + 4σ, we obtain two linearly independent solutions: ψ+ = rγ+ and ψ− = rγ− , where γ± = 1 2 [ 2−N ± √ (N − 2)(N − 10− 4σ) ] < 0, which are roots of the quadratic equation γ2 + (N − 2)γ + (2 + σ)(N − 2) = 0. If N = 10 + 4σ, we obtain two linearly independent solutions: ψ̃+ = r−4−2σ and ψ̃− = r−4−2σ log ( r 4 + 2σ ) . Let ψ1 and ψ̃1 be the first eigenfunctions of operate A in the cases N > 10 + 4σ and N = 10 + 4σ, respectively. As shown in [13, 17], we find, if N > 10 + 4σ, ψ1 = arγ+ ( 1 + o(1) ) , r → 0, (2.27) and if N = 10 + 4σ, ψ̃1 = ãr−4−2σ ( 1 + o(1) ) , r → 0, (2.28) where a = a(σ,N) > 0, ã = ã(σ,N) > 0. We state some properties of symmetric Sturm-Liouville operator A. The proof can be referred to [13, Lemma 4.1] and [17, Lemma 2.2]. Lemma 2.6. The operator A defined in (2.25) satisfies the following properties: (i) If N > 10 + 4σ, the operator A admits a unique self-adjoint Friedrichs ex- tension which is positive definite with a purely discrete spectrum. Moreover, the first eigenvalue is strictly positive and satisfies λ1 ≥ m = µ1 N − (10 + 4σ) N − 2 > 0, where µ1 > 0 is the first eigenvalue of −∆ in BR. (ii) If N = 10 + 4σ, the operator A admits a unique self-adjoint Friedrichs extension with a purely discrete spectrum of simple eigenvalues σ(A) = {· · · < λ̃2 < λ̃1 < 0}. Moreover, the orthonormal set of eigenfunctions {ψ̃k} for A is complete. 10 C. CHANG, Z. ZHANG EJDE-2022/42 2.3.2. Outer analysis. The following result gives the asymptotic behavior in the outer region, which is proved as in [13, Lemma 5.1]. Theorem 2.7. Assume N > 10 + 4σ. Let λ1 and ψ1 be the first eigenpair of operator A. Then there exists a positive constant C0 = C0(u0) such that as t→∞, w(r, t) = C0e −λ1tψ1(r) ( 1 + o(1) ) uniformly in {δ ≤ |x| ≤ R}, (2.29) where δ > 0. 2.4. Matching process. Combining Subsections 2.2 and 2.3, we obtain the desired result by matching the asymptotic behaviors of solutions in the inner and outer regions. Proof of Theorem 1.1. The proof is divided into three steps. In step 1, we observe the formal matching expansion of the global solution for (1.1). In steps 2 and 3, we give the analytic proof. Step 2 gives the estimate of upper bound. Step 3 gives the estimate of the lower bound. Step 1: α(t) ≈ (2+σ)λ1 |γ+| t + O(1) as t → ∞. Set r = δ with δ � 1. By (2.13) and (2.9), the function w = Us − u satisfies w(δ, t) ≈ b0e α(t)γ+ 2+σ δγ+ , t→∞. (2.30) Substituting (2.27) into (2.29), we obtain w(δ, t) = aC0e −λ1tδγ+ ( 1 + o(1) ) , t→∞. (2.31) Combining (2.30) and (2.31), we deduce α(t) = ‖u(x, t)‖∞ ≈ (2 + σ)λ1 |γ+| t+O(1), t→∞. Thus (1.11) is valid. Step 2: α(t) ≤ (2+σ)λ1 |γ+| t + O(1) as t → ∞. For a fixed positive r � 1, by (2.23), (2.8), and (2.9), we obtain u(r, t) ≥ Us(r)− b0e α(t)γ+ 2+σ rγ+ ( 1 + o(1) ) , t→∞. (2.32) It follows from (2.31) and (2.32) that e α(t)γ+ 2+σ ≥ aC0 b0 e−λ1t ( 1 + o(1) ) , t→∞. (2.33) Based on the continuity and positivity of ex, there exists C ′ > 0 such that aC0 b0 = eC ′ . By (2.33), we find α(t) ≤ (2 + σ)λ1 |γ+| t+ C ′(2 + σ) γ+ , t→∞. Therefore, α(t) ≤ (2 + σ)λ1 |γ+| t+O(1), t→∞. (2.34) Step 3: α(t) ≥ (2+σ)λ1 |γ+| t + O(1) as t → ∞. A direct calculation shows that w(r, t) = C0e −λ1(t−T )ψ1(r) is a supersolution of (2.24), where we fix T � 1 such EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 11 that w0(r) = Us(r)−u0(r) ≤ w(r, 0). Hence, u(r, t) = Us(r)−w(r, t) is a subsolution of (1.1), i.e. u(r, t) ≥ u(r, t) in BR × R+. By the monotonicity of u(r, t) with respect to r, we deduce for t ≥ T , α(t) = sup r u(r, t) ≥ sup r u(r, t) ≡ (2 + σ)λ1 |γ+| t+O(1), (2.35) where the supremum is attained at r ≈ [ σ+2 aC0|γ+| ] 1 γ+ e λ1t γ+ . � 3. The case N = 10 + 4σ This section concerns the asymptotic expansion of solutions for (1.1) in the critical case N = 10 + 4σ. The estimates in the inner and outer regions are similar to the ones in the case N > 10 + 4σ. The estimate in the inner region is given by Theorem 2.3, we give the estimate in the outer region as follows. The proof can be referred to [17, Lemma 2.4]. Lemma 3.1. Let N = 10 + 4σ. Then there exists a constant C0 > 0 such that, for sufficiently large t, w(r, t) ≤ min{Us(r), C0e λ̃1tψ̃1(r)}. (3.1) Remark 3.2. Using the proof of (2.34), we cannot deduce the estimate of upper bound of α(t). In fact, by (2.23), (2.8), and (2.10), we have for r � 1, u(r, t) ≥ Us(r)− b0e−2α(t)r−4−2σ log ( re α(t) 2+σ )( 1 + o(1) ) , t→∞. It follows from Lemma 3.1 and (2.28) that w(r, t) ≤ C0e λ̃1tψ̃1(r) = ãC0e λ̃1tr−4−2σ ( 1 + o(1) ) , t→∞. Then b0e −2α(t) log ( re α(t) 2+σ ) ≥ ãC0e λ̃1t ( 1 + o(1) ) , t→∞. Clearly, there are no terms which matches the term log(re α(t) 2+σ ). The following lemma presents the estimate of lower bound of α(t). Lemma 3.3. Suppose N = 10 + 4σ. Then α(t) defined in (2.11) satisfies α(t) ≥ |λ̃1| 2 t+ C1, t→∞, (3.2) where C1 > 0, λ̃1 < 0 is the first eigenvalue of operator A defined in (2.25). Proof. The proof is similar to (2.35). By (3.1), we have for sufficiently large T , u(r, t) ≥ Us(r)− C0e λ̃1(t−T )ψ̃1(r) in BR × (T,∞), where λ̃1 < 0 and ψ̃1(r) are the first eigenpair of operator A. Therefore, α(t) ≥ max r { log (2 + σ)(8 + 4σ) r2+σ − C0e λ̃1tψ̃1(r) } , where the supermum is attained at r ≈ e λ̃1t 4+2σ . � 12 C. CHANG, Z. ZHANG EJDE-2022/42 From the later calculation, we obtain that λ̃1 = − π2 1 (4+2σ) 4 2+σ , where π1 is the first zero of the zeroth-order Bessel’s function, i.e. J0(π1) = 0. In other words, Lemma 3.3 gives the optimal coefficient of t in (1.12). Moreover, we find that the second term C1 on the right-hand side of (3.2) is not optimal, and it turns out to be a logarithmically growing function in the later proof. 3.1. The inner problem. As shown in Sub-subsection 2.2.2, we introduce the rescaled function Φ0, which satisfies u(r, t) = α(t) + Φ0(ξ, t), ξ = re α(t) 2+σ , where α(t) = supr u(r, t) = u(0, t). Recalling the proof of Theorem 2.3, we find that the leading-order term of expansion of Φ0 is quasi-steady as t → ∞, and Φ0 satisfies Φ′′0 + N − 1 ξ Φ′0 + |ξ|σeΦ0 = 0, ξ ∈ (0, R), Φ0(0) = Φ′0(0) = 0. Indeed, Φ0 corresponds to U0 in Sub-subsection 2.2.1. Set Φ0 = log(2 + σ)− (2 + σ) log Ψ0. A simple calculation implies that Ψ′′0Ψ1+σ 0 − (Ψ′0)2Ψσ 0 + N − 1 ξ Ψ′0Ψ1+σ 0 = ξσ, (3.3) which is a nonlinear differential equation of order (2 + σ). Set Ψ0(ξ) ∼ ξ (N − 2) 1 2+σ + Ω(ξ), ξ →∞, (3.4) where Ω(ξ)→ 0 as ξ →∞. Substituting (3.4) into (3.3), we obtain 0 = Ω′′ [ ξ (N − 2) 1 2+σ + Ω ]1+σ − [ 1 (N − 2) 1 2+σ + Ω′ ]2[ ξ (N − 2) 1 2+σ + Ω ]σ + N − 1 ξ [ 1 (N − 2) 1 2+σ + Ω′ ][ ξ (N − 2) 1 2+σ + Ω ]1+σ − ξσ. Using Taylor expansion on the term [ ξ (N−2) 1 2+σ + Ω]β at the point ξ (N−2) 1 2+σ , we deduce that 0 = Ω′′ [( ξ (N − 2) 1 2+σ )1+σ + (1 + σ) ( ξ (N − 2) 1 2+σ )σ Ω + o(Ω) ] − [( 1 (N − 2) 1 2+σ )2 + (Ω′)2 + 2Ω′ (N − 2) 1 2+σ ] × [( ξ (N − 2) 1 2+σ )σ + σ ( ξ (N − 2) 1 2+σ )−1+σ Ω + o(Ω) ] + N − 1 ξ [ 1 (N − 2) 1 2+σ + Ω′ ] × [( ξ (N − 2) 1 2+σ )1+σ + (1 + σ) ( ξ (N − 2) 1 2+σ )σ Ω + o(Ω) ] − ξσ. (3.5) EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 13 By sorting out the terms with Ω′′, Ω′ and Ω, we rewrite (3.5) in the form 0 = [ ξ (N − 2) 1 2+σ ]1+σ Ω′′ + [ ξ (N − 2) 1 2+σ ]σ N − 3 (N − 2) 1 2+σ Ω′ + Nσ − 2σ +N − 1 (N − 2) 2 2+σ [ ξ (N − 2) 1 2+σ ]−1+σ Ω−G(ξ,N, σ), (3.6) where G(ξ,N, σ) = −Ω′′ [ (1 + σ) ( ξ (N − 2) 1 2+σ )σ Ω + o(Ω) ] + ( 1 (N − 2) 1 2+σ )2[( ξ (N − 2) 1 2+σ )σ + o(Ω) ] + (Ω′)2 [( ξ (N − 2) 1 2+σ )σ + σ ( ξ (N − 2) 1 2+σ )−1+σ Ω + o(Ω) ] + 2Ω′ (N − 2) 1 2+σ [ σ ( ξ (N − 2) 1 2+σ )−1+σ Ω + o(Ω) ] − N − 1 ξ 1 (N − 2) 1 2+σ [( ξ (N − 2) 1 2+σ )1+σ + o(Ω) ] − N − 1 ξ Ω′ [ (1 + σ) ( ξ (N − 2) 1 2+σ )σ Ω + o(Ω) ] + ξσ. (3.7) To solve Ω(ξ), we consider the homogeneous equation of (3.6): ξ2Ω′′ + ξ(N − 3)Ω′ + (Nσ − 2σ +N − 1)Ω = 0. If N > 10 + 4σ, Ω(ξ) = C1ξ q+ + C2ξ q− , (3.8) where C1, C2 > 0, and q± = 1 2 [ 4−N ± √ (N − 2)(N − 10− 4σ) ] , which are the roots of the quadratic equation q2 + (N − 4)q + (Nσ − 2σ +N − 1) = 0. If N = 10 + 4σ, Ω(ξ) = C3ξ −3−2σ log ξ + C4ξ −3−2σ, (3.9) where C3, C4 > 0. Next, we consider the case N = 10 + 4σ. Since lim ξ→∞ ξ−3−2σ log ξ = lim ξ→∞ 1 (3 + 2σ)ξ3+2σ = 0 and lim ξ→∞ ξ−3−2σ log ξ ξ−3−2σ = lim ξ→∞ log ξ =∞, we find that the convergence rate of term ξ−3−2σ to 0 is faster than that of term ξ−3−2σ log ξ to 0. Hence, Ω(ξ) = C3ξ −3−2σ log ξ, ξ →∞. (3.10) We compute Ω′(ξ) = −(3 + 2σ)C3ξ −4−2σ log ξ + C3ξ −4−2σ = −(3 + 2σ)C3ξ −4−2σ log ξ, (3.11) 14 C. CHANG, Z. ZHANG EJDE-2022/42 and Ω′′(ξ) = (3 + 2σ)(4 + 2σ)C3ξ −5−2σ log ξ − (7 + 4σ)C3ξ −5−2σ = (3 + 2σ)(4 + 2σ)C3ξ −5−2σ log ξ. (3.12) Substituting (3.10)-(3.12) into (3.7), we obtain G(ξ,N, σ) = N − 1 + (N − 4)σ (N − 2) σ 2+σ ξ−8−3σ(log ξ)2+ σ (N − 2) −1+σ 2+σ ξ−12−5σ(log ξ)3+o(1). Since lim ξ→∞ ξ−8−3σ(log ξ)2 = lim ξ→∞ 2 (8 + 3σ)2ξ8+3σ = 0, lim ξ→∞ ξ−12−5σ(log ξ)3 = lim ξ→∞ 6 (12 + 5σ)3ξ12+5σ = 0, we have G(ξ,N, σ)→ 0, ξ →∞. (3.13) Thus, Ω(ξ) can be given by (3.8) and (3.9). By (3.4), (3.9), and the fact log(1 + x) ∼ x as x→ 0, we have as ξ →∞, Φ0(ξ) = log(2 + σ)− (2 + σ) log Ψ0 ∼ log(2 + σ)− log [ ξ (8 + 4σ) 1 2+σ + C3ξ −3−2σ log ξ + C4ξ −3−2σ ]2+σ = log(2 + σ)− log [ ξ (8 + 4σ) 1 2+σ ]2+σ − (2 + σ) log [ 1 + (8 + 4σ) 1 2+σ ( C3ξ −4−2σ log ξ + C4ξ −4−2σ )] ∼ log (4 + 2σ)2 ξ2+σ − (2 + σ)(8 + 4σ) 1 2+σ ξ−4−2σ(C3 log ξ + C4). Therefore, Φ0(ξ) ∼ log (4 + 2σ)2 ξ2+σ − ξ−4−2σ(A0 log ξ +B0), ξ →∞, (3.14) where A0 = (2 + σ)(8 + 4σ) 1 2+σC3 and B0 = (2 + σ)(8 + 4σ) 1 2+σC4. For the case N > 10 + 4σ, we can also get an accurate estimate of lower order term, which is not needed in this paper. Notice that the term e µγ+ 2+σ rγ+ in (2.9) is sufficient to match the resulting terms in the outer analysis. 3.2. The outer problem. The leading-order term of expansion of u in the outer region is log (4+2σ)2 r2+σ as t → ∞, which is given by the leading-order term in (3.14). In order to study the correction term, we give the behavior of u in the form of u(r, t) ∼ log (4 + 2σ)2 r2+σ − v(r, t), t→∞, EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 15 where v corresponds to w in Subsection 2.1. The linearized problem of v is given by vt = vrr + 9 + 4σ r vr + (4 + 2σ)2 r2 v in ( 0, (4 + 2σ) 1 2+σ ) × R+, v((4 + 2σ) 1 2+σ , t) ≡ 0. (3.15) To match logarithmic term in (3.14), we set v(r, t) = e − π2t (4+2σ) 4 2+σ [ β(t)φ0(r) + β̇(t)φ1(r) + . . . ] , (3.16) where π and β(t) will be chosen later. Substituting(3.16) into (3.15), we obtain φ′′0(r) + 9 + 4σ r φ′0(r) + [(4 + 2σ r )2 + π2 (4 + 2σ) 4 2+σ ] φ0(r) = 0, φ′′1(r) + 9 + 4σ r φ′1(r) + [(4 + 2σ r )2 + π2 (4 + 2σ) 4 2+σ ] φ1(r) = φ0(r). Let φ0(r) = r−4−2σP0(r), φ1(r) = r−4−2σP1(r). Then P ′′0 (r) + 1 r P ′0(r) + π2 (4 + 2σ) 4 2+σ P0(r) = 0, (3.17) P ′′1 (r) + 1 r P ′1(r) + π2 (4 + 2σ) 4 2+σ P1(r) = P0(r). (3.18) Equation (3.17) is the zeroth-order Bessel’s equation and P0(r) = J0 ( πr (4+2σ) 2 2+σ ) . On the other hand, by (3.14), we have v(r, t) ∼ A0 2 + σ α(t)e−2α(t)r−4−2σ + e−2α(t) ( A0r −4−2σ log r +B0r −4−2σ ) , (3.19) as t→∞. To match (3.16) and (3.19), it suffices to show that β(t)e − π2 1t (4+2σ) 4 2+σ ∼ A0 2 + σ α(t)e−2α(t), t→∞. (3.20) We choose π1 to be the first zero of zeroth-order Bessel’s function, and J0(π1) = 0. It follows from (3.17) and (3.18) that r(P ′1P0 + P ′0P1) = − ∫ (4+2σ) 2 2+σ s rP 2 0 (s)dr = − (4 + 2σ) 4 2+σ 2 J2 1 (π1) + r2 2 [ J2 0 ( π1r (4 + 2σ) 2 2+σ ) + J2 1 ( π1r (4 + 2σ) 2 2+σ )] . (3.21) Since P0(0) is a constant, we deduce form (3.21) that P1(r) = − (4 + 2σ) 4 2+σ 2 J2 1 (π1) log r +O(1), r → 0. 16 C. CHANG, Z. ZHANG EJDE-2022/42 By (3.16) and (3.19), we require − (4 + 2σ) 4 2+σ 2 J2 1 (π1)β̇(t)e − π2 1t (4+2σ) 4 2+σ ∼ A0e −2α(t). (3.22) Combining (3.20) and (3.22), we find α(t) = π2 1t 2(4 + 2σ) 4 2+σ + α1(t), t→∞. (3.23) The function β(t) is determined by β(t) ∼ A0π 2t (4 + 2σ) 6+σ 2+σ e−2α1(t), β̇(t) ∼ − 2A0 (4 + 2σ) 4 2+σ J2 1 (π1) e−2α1(t). Therefore, β(t) ∼ β∞t − 8+4σ π2 1J 2 1 (π1) t→∞, where β∞ > 0 depends only on u0. Then α1(t) ∼ 1 2 [ 1 + 8 + 4σ π2 1J 2 1 (π1) ] log t+ 1 2 log [ A0π 2t (4 + 2σ) 6+σ 2+σ ] , t→∞. (3.24) We deduce from (3.23) and (3.24) that, as t→∞, u(0, t) ∼ π2 1t 2(4 + 2σ) 4 2+σ + 1 2 [ 1 + 8 + 4σ π2 1J 2 1 (π1) ] log t+ 1 2 log [ A0π 2t (4 + 2σ) 6+σ 2+σ ] . It follows that u(0, t) grows linearly as t→∞ with a logarithmic correction term. Remark 3.4. Combining the estimates in the inner and outer regions, we derive the widths of the inner layer, which are estimated as O(e − λ1 |γ+| t ) if N > 10+4σ and O ( e − π2 1t (4+2σ) 6+σ 2+σ t− 1 2+σ ) if N = 10 + 4σ. Indeed, since inner analysis is studied on compact set {ξ = re α(t) 2+σ ≤ C} with C > 0, we have that r = O ( e− α(t) 2+σ ) . It follows from Theorem 1.1 that r = O ( e− α(t) 2+σ ) = O ( e − 1 2+σ [ (2+σ)λ1 |γ+| t+O(1)] ) = O ( e − λ1 |γ+| t ) . It follows from Theorem 1.2 that r = O ( e− α(t) 2+σ ) = O ( e − 1 2+σ [ π2 1t 2(4+2σ) 4 2+σ +O(log t)]) = O ( e − π2 1t (4+2σ) 6+σ 2+σ t− 1 2+σ ) . Acknowledgments. This work was partially supported by the National Natural Science Foundation of China (No. 12071044). References [1] N. D. Alikakos, P. W. Bates, C. P. Grant; Blow up for a diffusion-advection equation, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 181–190. [2] J. M. Arrieta, A. B. Rodriguez, Ph. Souplet; Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 1–15. [3] M. Badiale, E. Serra; Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453–467. [4] E. M. Barboza, O. H. Miyagaki, F. R. Pereira, C. R. Santana; Hénon equation with nolin- earities involving Sobolev critical growth in H1 0,rad(B1), Electron. J. Differential Equations, Vol. 2021 (2021), No. 20, 1–18. EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 17 [5] J. Bebernes, S. Bricher; Final time blowup profiles for semilinear parabolic equations via center manifoldtheory, SIAM J. Math. Anal., 23 (1992), 852–869. [6] H. Brezis, J. L. Vázquez; Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443–469. [7] D. M. Cao, S. J. Peng; The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1–17. [8] C. H. Chang, Q. C. Ju, Z. C. Zhang; Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity, Discrete Contin. Dyn. Syst., 40 (2020), 5991–6014. [9] H. Y. Chen, D. Ye, F. Zhou; On Gaussian curvature equation in R2 with prescribed nonpos- itive curvature, Discrete Contin. Dyn. Syst., 40 (2020), 3201–3214. [10] G. Conner, C. P. Grant; Asymptotics of blowup for a convection-diffusion equation with conservation, Differential Integral Equations, 9 (1996), 719–728. [11] M. G. Crandall, P. H. Rabinowitz; Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207–218. [12] T. D’Aprile; Sign-changing blow-up solutions for Hénon type elliptic equations with expo- nential nonlinearity, J. Funct. Anal., 268 (2015), 2067–2101. [13] J. W. Dold, V. A. Galaktionov, A. A. Lacey, J. L. Vázquez; Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 663–687. [14] S. Filippas, R. V. Kohn; Refined asymptotics for the blow-up of ut = ∆u+ up, Comm. Pure Appl. Math., 45 (1992), 821–869. [15] V. A. Galaktionov, J. R. King; Fast diffusion equation with critical Sobolev exponent in a ball, Nonlinearity, 15 (2002), 173–188. [16] V. A. Galaktionov, J. R. King; Composite structure of global unbounded solutions of non- linear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199–233. [17] V. A. Galaktionov, J. R. King; Stabilization to a singular steady state for the Frank- Kamenetskii equation in a critical dimension, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 777–787. [18] I. M. Gel’fand; Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295–381. [19] N. Ghoussoub, Y. J. Guo; On the partial differential equations of electrostatic MEMS devices. II. Dynamic case, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 115–145. [20] Y. Giga, R. V. Kohn; Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297–319. [21] B. Gidas, J. Spruck; Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525–598. [22] Y. J. Guo; On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior, J. Differential Equations, 244 (2008), 2277–2309. [23] Y. J. Guo; Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Differential Equations, 245 (2008), 809–844. [24] Z. M. Guo, J. Y. Li, F. S. Wan; Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics, Comm. Partial Differential Equations, 45 (2020), 1647–1681. [25] M. Hénon; Numerical experiments on the stability oh spherical stellar systems, Astron. As- trophys., 24 (1973), 229–238. [26] M. A. Herrero, J. J. L. Velázquez; Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17 (1992), 205–219. [27] M. A. Herrero, J. J. L. Velázquez; Flat blow-up in one-dimensional semilinear heat equations, Differential Integral Equations, 5 (1992), 973–997. [28] M. A. Herrero, J. J. L. Velázquez; Generic behaviour of one-dimensional blow up patterns, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 381–450. [29] M. A. Herrero, J. J. L. Velázquez; Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 131–189. [30] D. D. Joseph, T. S. Lundgren; Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972), 241–269. 18 C. CHANG, Z. ZHANG EJDE-2022/42 [31] Q. C. Ju, H. L. Li, Y. Li, S. Jiang; Quasi-neutral limit of the two-fluid Euler-Poisson, Comm. Pure Appl. Anal., 9 (2010), 1577–1590. [32] N. I. Kavallaris, Ph. Souplet; Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008), 1852–1881. [33] A. A. Lacey, D. Tzanetis; Global existence and convergence to a singular steady state for a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 289–305. [34] Y. X. Li; Stabilization towards the steady state for a viscous Hamilton-Jacobi equation, Commum. Pure Appl. Anal., 8 (2009), 1917–1924. [35] S. J. Li, S. J. Peng; Asymptotic behavior on the Hénon equation with supercritical exponent, Sci. China Ser. A, 52 (2009), 2185–2194. [36] Y. X. Li, Ph. Souplet; Single-point gradient blow-up on the boundary for diffusive Hamilton- Jacobi equations in planar domains, Comm. Math. Phys., 293 (2010), 499–517. [37] T. Matukuma; Sur la dynamique des amas globulaires stellaires, Proc. Imp. Acad. Tokyo, 6 (1930), 133–136. [38] F. Merle, H. Zaag; Optimal estimates for blowup rate and behavior for nonlinear heat equa- tions, Comm. Pure Appl. Math., 51 (1998), 139–196. [39] W. M. Ni; A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801–807. [40] I. Peral, J. L. Vázquez; On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal., 129 (1995), 201– 224. [41] M. del Pino, M. Kowalczyk, M. Musso; Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47–81. [42] A. Pistoia, E. Serra; Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75–97. [43] A. Porretta, Ph. Souplet; The profile of boundary gradient blowup for the diffusive Hamilton- Jacobi equation, Int. Math. Res. Not. IMRN, 2017, 5260–5301. [44] A. Porretta, Ph. Souplet; Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1913– 1923. [45] E. Serra; Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301–326. [46] D. Smets, M. Willem; Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57–75. [47] D. Smets, M. Willem, J. B. Su; Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467–480. [48] Ph. Souplet; Gradient blow-up for multidimensional nonlinear parabolic equations with gen- eral boundary conditions, Differential Integral Equations, 15 (2002), 237–256. [49] Ph. Souplet, S. Tayachi; Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math., 88 (2001), 135–154. [50] Ph. Souplet, J. L. Vázquez; Stabilization towards a singular steady state with gradient blow- up for a diffusion-convection problem, Discrete Contin. Dyn. Syst., 14 (2006), 221–234. [51] Ph. Souplet, Q. S. Zhang; Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math., 99 (2006), 355–396. [52] S. Takasi; Blowup in infinite time of radial solutions for a parabolic-elliptic system in high- dimensional Euclidean spaces, Nonlinear Anal., 70 (2009), 2549–2562. [53] M. Troyanov; Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., 324 (1991), 793–821. [54] Z. C. Zhang, B. Hu; Gradient blowup rate for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 767–779. [55] Z. C. Zhang, B. Hu; Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594–4601. [56] Z. C. Zhang, Y. Li; Global existence and gradient blowup of solutions for a semilinear par- abolic equation with exponential source, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3019–3029. [57] Z. C. Zhang, Y. Li; Classification of blowup solutions for a parabolic p-Laplacian equation with nonlinear gradient terms, J. Math. Anal. Appl., 436 (2016), 1266–1283. EJDE-2022/42 ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS 19 Caihong Chang School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China Email address: caihong666@stu.xjtu.edu.cn Zhengce Zhang (corresponding author) School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China Email address: zhangzc@mail.xjtu.edu.cn 1. Introduction 1.1. Background 1.2. Known results 1.3. Main results 2. The case N>10+4 2.1. Preliminary estimates 2.2. Asymptotic behavior in the inner region 2.3. Asymptotic behavior in the outer region 2.4. Matching process 3. The case N=10+4 3.1. The inner problem 3.2. The outer problem Acknowledgments References