Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplace operator and nonlinear sources

The aim of this paper was to study vanishing and blowing-up properties of the solutions to a homogeneous initial Dirichlet problem of a nonlinear diffusion equation involving the p(x)-Laplace operator and a nonlinear source. The authors point out that the results obtained are not trivial generalizations of similar problems in the case of constant exponent because the variable exponent p(x) brings some essential difficulties such as the failure of upper and lower solution method and scaling technique, the existence of a gap between the modular and the norm. To overcome these difficulties, the authors have to improve the regularity of solutions, to construct a new control functional and apply suitable embedding theorems to prove the blowing-up property of the solutions. In addition, the authors utilize an energy estimate method and a comparison principle for ODE to prove that the solution vanishes in finite time. At the same time, the critical extinction exponents and an extinction rate estimate to the solutions are also obtained.