Abstract
This paper deals with blow-up phenomena for an initial boundary value problem of a nonlocal quasilinear parabolic equation with time-dependent coefficients in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficients and nonlinearities to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, upper and lower bounds of $t^{\ast}$ are obtained under suitable measure in high-dimensional spaces. Finally, some application examples are presented.
Highlights
In the past decades, there have been many works dealing with existence and nonexistence of global solutions, blow-up of solutions, bounds for blow-up time, blow-up rates, blow-up sets, and asymptotic behavior of the solutions to nonlinear parabolic equations. we refer the reader to the monographs [1, 20, 21] as well as to the survey paper [10, 14]
Under some conditions on k(t) and f (u), they obtained some sufficient conditions for existence of the global solution and upper bound for the blow-up time
− k(t)f (u), xj (x, t) ∈ Ω × (0, t∗), i,j=1 under nonlinear boundary flux, where Ω ⊂ RN (N ≥ 2) is a bounded star-shaped region with smooth boundary ∂Ω. They established some conditions on initial data which guarantee blow-up or global existence of the solutions and derived an upper bound of the blow-up time
Summary
There have been many works dealing with existence and nonexistence of global solutions, blow-up of solutions, bounds for blow-up time, blow-up rates, blow-up sets, and asymptotic behavior of the solutions to nonlinear parabolic equations. we refer the reader to the monographs [1, 20, 21] as well as to the survey paper [10, 14]. Under some conditions on k(t) and f (u), they obtained some sufficient conditions for existence of the global solution and upper bound for the blow-up time.
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