Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source

Consider a class of integrodifferential of parabolic equations involving variable source with Dirichlet boundary condition \begin{equation*} u_{t}=\Delta u-\int _{0}^{t}g\left(t-s\right) \Delta u\left(x,s\right) \mathrm{d} s+|u| ^{p(x) -2}u. \end{equation*} By means energy methods, we obtain a lower bound for blow-up time of the solution if blow-up occurs. Furthermore, assuming the initial energy is negative we establish a new blow-up criterion and give an upper bound for blow-up time of the solution

Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source

Consider a class of integrodifferential of parabolic equations involving variable source with Dirichlet boundary condition \begin{equation*} u_{t}=\Delta u-\int _{0}^{t}g\left(t-s\right) \Delta u\left(x,s\right) \mathrm{d} s+|u| ^{p(x) -2}u. \end{equation*} By means energy methods, we obtain a lower bound for blow-up time of the solution if blow-up occurs. Furthermore, assuming the initial energy is negative we establish a new blow-up criterion and give an upper bound for blow-up time of the solution