In this work, we investigate the following Kirchhoff-type equation with variable exponent nonlinearities u_{tt}-M(‖∇u‖²)△u+|u_{t}|^{p(x)-2}u_{t}=|u|^{q(x)-2}u. We proved the blow up of solutions in finite time by using modified energy functional method.

In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called m...

In this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for the heat equation, ut = ∆u, in a bounded smooth domain, with a nonlinear flux boundary condition at the boundary, ∂u ∂η = up. We focus in the behaviour of blowing up solutions. First we prove that every numerical solution blows up in finite time if and only if p > 1 and that the numerical blow-up time ...