On the critical behavior for time-fractional pseudo-parabolic-type equations with combined nonlinearities

Abstract

We are concerned with the existence and nonexistence of global weak solutions for a certain class of time-fractional inhomogeneous pseudo-parabolic-type equations involving a nonlinearity of the form |u|^{p}+iota |nabla u|^{q}, where p,q>1, and iota geq 0 is a constant. The cases iota =0 and iota >0 are discussed separately. For each case, the critical exponent in the Fujita sense is obtained. We point out two interesting phenomena. First, the obtained critical exponents are independent of the fractional orders of the time derivative. Secondly, in the case iota >0, we show that the gradient term induces a discontinuity phenomenon of the critical exponent.