id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-140	Wang, Ru; Chang, Xiaojun	Existence of global solutions and blow-up for p-Laplacian parabolic equations with logarithmic nonlinearity on metric graphs	2022	18	.pdf	application/pdf	7409	497	83	Integrating on (0, t) with t ∈ (0, T ), by 1 < p < 2 it follows that 1 2 ∫ G |w(t)|2dx ≤ ∫ t 0 ∫ G ((p− 1) log |w̃(s)|+ 1)|w̃(s)|p−2|w(s)|2 dx ds ≤ C ∫ t 0 ∫ G |w(s)|2 dx ds for some constant C > 0 independent of u1(t) and u2(t). A function u := u(x, t) is called a weak solution of problem (1.1) on G × (0, T∗), if u ∈ L∞(0, T∗;W 1,p(G)) with ut ∈ L2(0, T∗;L 2(G)) satisfies (1.1) in the distribution sense, i.e.,∫ G utvdx+ ∫ G |u′|p−2u′v′dx+ ∫ G |u|p−2uvdx = ∫ G |u|p−2uv log |u|dx, for all v ∈W 1,p(G), a.e. t ∈ (0, T∗), where u(x, 0) = u0(x) ∈ X0. 4 R. WANG, X. J. CHANG EJDE-2022/51 Our first result is about the existence of a local solution.	cache/ejde-140.pdf	txt/ejde-140.txt