Blow-up phenomena for a pseudo-parabolic system with variable exponents
Qi Qi, Yujuan Chen, Qingshan Wang · Electronic journal of qualitative theory of differential equations · 2017
In this paper, we consider a pseudo-parabolic system with nonlinearities of variable exponent type \begin{align*} \begin{cases} u_t-\Delta u_t-\operatorname{div}(| abla u|^{m(x)-2} abla u)=|uv|^{p(x)-2}uv^2 & \mbox{in}\ \Omega\times(0,T),\\ v_t-\Delta v_t-\operatorname{div}(| abla v|^{n(x)-2} abla v)=|uv|^{p(x)-2}u^2v & \mbox{in}\ \Omega\times(0,T) \end{cases} \end{align*} associated with initial and Dirichlet boundary conditions, where the variable exponents $p(\cdot)$, $m(\cdot)$, $n(\cdot)$ are continuous functions on $\overline{\Omega}$. We obtain an upper bound and a lower bound for blow-up time if variable exponents $p(\cdot)$, $m(\cdot)$, $n(\cdot)$ and the initial data satisfy some conditions.