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

Rahmoune, Abita

Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 23-32.

Considérons une classe d’équations intégro-différentielles paraboliques comprenant une source variable et avec condition de Dirichlet au bord

$$u_t=\Delta u-{\int }_0^{t}g\left(t-s\right)\Delta u\left(x,s\right)\mathrm{d}s+{\left|u\right|}^{p\left(x\right)-2}u.$$

À l’aide des méthodes d’énergie nous obtenons une borne inférieure pour le temps où intervient une éventuelle explosion de la solution. De plus, en supposant que l’énergie initiale est négative nous établissons un nouveau critère pour l’explosion et nous donnons une borne supérieure pour le temps d’explosion de la solution.

Consider a class of integrodifferential of parabolic equations involving variable source with Dirichlet boundary condition

$$u_t=\Delta u-{\int }_0^{t}g\left(t-s\right)\Delta u\left(x,s\right)\mathrm{d}s+{\left|u\right|}^{p\left(x\right)-2}u.$$

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.

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