No 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 $|u|^{p}+\iota |\nabla u|^{q}$ | u | p + ι | ∇ u | q , where $p,q>1$ p , q > 1 , and $\iota \geq 0$ ι ≥ 0 is a constant. The cases $\iota =0$ ι = 0 and $\iota >0$ ι > 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 $\iota >0$ ι > 0 , we show that the gradient term induces a discontinuity phenomenon of the critical exponent.

We first consider the damped wave inequality ∂2u∂t2−∂2u∂x2+∂u∂t≥xσ|u|p,t>0,x∈(0,L), where L>0, σ∈R, and p>1, under the Dirichlet boundary conditions (u(t,0),u(t,L))=(f(t),g(t)),t>0. We establish sufficient conditions depending on σ, p, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: g≡0 and g(t)=tγ, γ>−1. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality ∂αu∂tα−∂2u∂x2+∂βu∂tβ≥xσ|u|p,t>0,x∈(0,L), where α∈(1,2), β∈(0,1), and ∂τ∂tτ is the time-Caputo fractional derivative of order τ, τ∈{α,β}. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm.