Abstract
We first consider the damped wave inequality partial differential 2u partial differential t2- partial differential 2u partial differential x2+ partial differential u partial differential t & GE;x sigma|u|p,t > 0,x & ISIN;(0,L), where L > 0, sigma & ISIN;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 sigma, 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 & EQUIV;0 and g(t)=t gamma, gamma >-1. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality partial differential alpha u partial differential t alpha- partial differential 2u partial differential x2+ partial differential beta u partial differential t beta & GE;x sigma|u|p,t > 0,x & ISIN;(0,L), where alpha & ISIN;(1,2), beta & ISIN;(0,1), and partial differential tau partial differential t tau is the time-Caputo fractional derivative of order tau, tau & ISIN;{alpha,beta}. 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.