Let \(d \in \{3,4,5,\ldots \}\) and \(\Omega \subset \mathbb {R}^d\) be open bounded with Lipschitz boundary. Let \(Q = \Omega \times (0,\infty )\) and \(p \in C({\overline{Q}})\) be such that \( 2< p^- \le p(\cdot ) \le p^+ < \frac{2^*}{2}+1, \) where \(2^*\) is the critical Sobolev exponent of 2, \( p^- := \mathop {\mathrm {ess\,inf}}\limits _{(x,t) \in Q} p(x,t) \quad \text {and}\quad p^+ := \mathop {\mathrm {ess\,sup}}\limits _{(x,t) \in Q} p(x,t). \) Consider the reaction-diffusion parabolic problem \( (P) \quad \left\{ \begin{array}{ll} \displaystyle \frac{u_t}{|x|^2} - \Delta u = k(t) \, |u|^{p(x,t)-2}u & (x,t) \in \Omega \times (0,T), \\ u(x,t) = 0, & (x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x), & x \in \Omega , \end{array}\right. \) where \(T > 0\) and \(0 \ne u_0 \in W^{1,2}_0(\Omega )\) . 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.
