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Global existence and blow-up analysis for parabolic equations with nonlocal source and nonlinear boundary conditions

Boundary Value Problems, ISSN: 1687-2770, Vol: 2020, Issue: 1
2020
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Article Description

We investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions: {(g(u))t=∑i,j=1N(aij(x)uxi)xj+γ1um(∫Duldx)p−γ2urin D×(0,t∗),∑i,j=1Naij(x)uxiνj=h(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾, where p and γ are some nonnegative constants, m, l, γ, and r are some positive constants, D⊂ R (N≥ 2) is a bounded convex region with smooth boundary ∂D. By making use of differential inequality technique and the embedding theorems in Sobolev spaces and constructing some auxiliary functions, we obtain a criterion to guarantee the global existence of the solution and a criterion to ensure that the solution blows up in finite time. Furthermore, an upper bound and a lower bound for the blow-up time are obtained. Finally, some examples are given to illustrate the results in this paper.

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