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The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for ut=div(|∇u|p(x,t)−2∇u+b(x,t)∇u)+f(u)$u_{t}=\operatorname{div}(|\nabla u|^{p(x,t)-2}\nabla u+b(x,t)\nabla u)+f(u)$. By constructing suitable function spaces and applying the method of Galerkin’s approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable test-function, make integral estimates, and apply Gronwall’s inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that pt(x,t)$p_{t}(x,t)$ is only negative and integrable, which is weaker than most of the other papers required.

Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with p ( x , t ) $p(x,t)$ -Laplacian operator