Abstract
In this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.
Highlights
The nonexistence of global solutions is a nonlinear Liouville-type theorem, and we can prove some properties of solutions in a bounded domain, which is a kind of essential reflection of blowup or singularity theory as well; see [1]
L√ater, Glassy [3] showed the nonexistence of global solution for the critical value p = 1 +
The purpose of this paper is finding the influence of nonlocal singular source term and nonhomogeneous term on the nonexistence of nontrivial global weak solution by the test function method developed in [8, 9]
Summary
We consider the Cauchy problems for quasilinear hyperbolic homogeneous and nonhomogeneous inequalities with nonlocal singular source term of the forms, respectively, utt ≥ utt ≥. Where S∞ = RN × (0, ∞), m, q, r, s > 0, r(q + s) > max{1, m}(r + s), sσ α> , r the initial functions u0, u1 ∈ L1loc(RN ) are nonnegative, the weight function β is positive and singular at the origin, that is, there exist constants c > 0 and σ ∈ R+ such that β(x) ≥ c|x|–σ > 0, x ∈ RN \{0},. The nonhomogeneous term w ∈ L1loc(RN ) is nonnegative
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