Blow-up and blow-up-time estimates for a singular pseudo-parabolic equation with a space-time variable exponent

Abstract

Let $d\in \{3,4,5,\dots \}$ and $Ω\subset \Ri^d$ be open bounded with Lipschitz boundary. Let $Q=\Omega \times (0,\infty )$ and $p\in C(\overline{Q})$ be such that \[ 2 < p^- \le p(\cdot) \le p^+ < 2^* := \frac{2d}{d-2}, \] where $ p^- := \essinf_{(x,t) \in Q} p(x,t) $ and $ p^+ := \esssup_{(x,t) \in Q} p(x,t). $ Consider the reaction-diffusion parabolic problem \[ (P) \quad \left\{\begin{array}{ll} \displaystyle\frac{u_t}{|x|^2} - Δu = k(t) \, |u|^{p(x,t)-2}u & (x,t) \in Ω\times (0,T), u(x,t) = 0, & (x,t) \in \partial Ω\times (0,T), \smallskip u(x,0) = u_0(x), & x \in Ω, \end{array}\right. \] where $T>0$ and $0\ne u_0\in W_0^{1,2}(\Omega )$. We investigate the existence and uniqueness of a weak solution to $(P)$. The upper and lower bounds on the blow-up time of the weak solution are also considered.