Abstract
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.
Highlights
In this paper, we first consider the damped wave inequality∂ u − ∂ u + ∂u ≥ x σ |u| p, t > 0, x ∈ (0, L), ∂t ∂x2(u(t, 0), u(t, L)) = ( f (t), g(t)), t > 0,∂u (0, x ) = (u0 ( x ), u1 ( x )), x ∈ (0, L), u(0, x ), (1)where L > 0, σ ∈ R, and p > 1
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
We establish sufficient conditions depending on the initial values, the boundary conditions, p, and σ, under which
Summary
The investigation of the question of blow-up of solutions to initial boundary value problems for semilinear wave equations started in the 1970s. By means of the energy method, the author established sufficient conditions for the blow-up of solutions. For further blow-up results for nonlinear wave equations, obtained by means of the energy/concavity method, see e.g., [3,4,5,6,7,8,9,10,11] and the references therein. In all the above cited references, the blow-up results were obtained for sufficiently large initial data. Taking into consideration the boundedness of the domain as well as the boundary conditions, adequate test functions are used to obtain sufficient conditions for the nonexistence of global weak solutions to problems (1) and (2).
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