The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

Abstract

This paper deals with the critical exponents for the quasi-linear parabolic equations in R n and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n 3, σ > −2/n and p > max{1, 1 + σ }, we obtain that p c = n(1 + σ )/(n − 2) is the critical exponent of these equations. Furthermore, we prove that if max{1, 1 + σ } < p p c , then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f (x) and some initial data u 0 (x) if p > p c . Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n = 1, 2, σ > −1 and p > max{1, 1 + σ }.

Existence of solutions for an inhomogeneous fractional semilinear heat equation

“…On the other hand, the existence of solutions to nonlinear parabolic equations with inhomogeneous terms has been studied in many papers, see e.g. [2,3,4,14,15,16,17,18,25,26,27,28] and references therein. However, there are no results concerning the identification of the strongest spatial singularity of the inhomogeneous term for the existence of solutions.…”

Blow-up and global existence for semilinear parabolic systems with space-time forcing terms

“…Note that system (1.5) (with positive solutions) was invetigated by Zhang [15] in a non-compact complete Riemannian manifold. For other contributions related to inhomogeneous problems, see, for example [3,13,14] and the references therein.…”

Fujita Critical Curve for a Coupled Diffusion System With Inhomogeneous Neumann Boundary Conditions∗

“…Later, it was proved by Hayakawa [7] and Weissler [10] that p = p c belongs to the blow-up case. From then on, there have been a lot of works on the critical exponents of Fujita type for various nonlinear evolution equations and systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Among these, Zeng [12] investigated the blow-up theorems of Fujita type for the following inhomogeneous problems…”

“…Further, we prove that the case pq = (pq) c belongs to the blow-up case. The method we used is similar to [1,2,3,9,11,12].…”