Existence and non-existence of global solutions for a heat equation with degenerate coefficients

2022/02/21 by Ricardo Castillo, Castillo, Ricardo, Omar Guzmán-Rea +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2202.10493

Abstract

In this paper, we will study the following parabolic problem u_t - div(ω(x) ∇ u)= h(t) f(u) + l(t) g(u) with non-negative initial conditions pertaining to C_b(ℝ^N), where the weight ω is an appropriate function that belongs to the Munckenhoupt class A_{1 + (2)/(N)} and the functions f, g, h and l are non-negative and continuous. The main goal is to establish of global and non-global existence of non-negative solutions. In addition, to present the particular case when h(t) ∼ t^r ~~ (r>-1), l(t) ∼ t^s ~~ (s>-1), f(u) = u^p and g(u)= (1+u)[ln(1+u)]^p, we obtain both the so-called Fujita's exponent and the second critical exponent in the sense of Lee and Ni \citeLee-Ni. Our results extend those obtained by Fujishima et al. \citeFujish who worked when h(t)=1, l(t)=0 and f(u)=u^p .

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