Global well‐posedness and finite time blow‐up for a class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearity

Abstract

AbstractIn this paper, we consider the following class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearitywhere is a bounded domain with Lipschitz boundary, , , and is the critical exponent in the Sobolev inequality. First, via the Galerkin approximations, the existence of local solutions are obtained when . Next, by combining the potential well theory with the Nehari manifold, we establish the existence of global solutions when . Then, via the Pohozaev manifold, the existence of global solutions are obtained when . By virtue of a differential inequality technique, we prove that the local solutions blow‐up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we discuss the asymptotic behavior of solutions as time tends to infinity. Here, we point out that the main difficulty is the lack of logarithmic Sobolev inequality concerning fractionalp‐Laplacian.