Global Existence and Blow-up of Solutions to a Quasilinear Parabolic Equation with Nonlocal Source and Nonlinear Boundary Condition

Abstract

This paper investigates the behavior of positive solution to the following $p$-Laplacian equation \begin{align*}u_t - (|u_x|^{p-2}u_x)_x = \int_{0}^a u^{\alpha}(\xi,t)d\xi+ku^\beta(x,t),\quad (x,t)\in[0,a]\times(0,T)\end{align*}with nonlinear boundary condition $u_x|_{x=0}=0$, $u_x|_{x=a}=u^q|_{x=a}$, where $p\geq 2$, $\alpha, \beta, k,q>0$. The authors first get the local existence result by a regularization method. Then under appropriate hypotheses, the authors establish that positive weak solution either exists globally or blow up in finite time by using comparison principle.