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

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 |abla u|^q, where p,q >1 and $iota geq 0$ is a constant. The casesiota=0 and iota >0 are discussed separately. For each case, the critical exponent in the sense of Fujita is obtained. We point out two interesting phenomena. First, the obtained critical exponents are independent on the fractional orders of the time-derivative. Secondly, in the caseiota>0, we show that the gradient term induces a discontinuity phenomenon of the critical exponent.