Search results for "global weak solutions"
showing 2 items of 2 documents
On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain
2021
Abstract We study the wave inequality with a Hardy potential ∂ t t u − Δ u + λ | x | 2 u ≥ | u | p in ( 0 , ∞ ) × Ω , $$\begin{array}{} \displaystyle \partial_{tt}u-{\it\Delta} u+\frac{\lambda}{|x|^2}u\geq |u|^p\quad \mbox{in } (0,\infty)\times {\it\Omega}, \end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − N − 2 2 2 $\begin{array}{} \displaystyle \left(\frac{N-2}{2}\right)^2 \end{array}$ , under the inhomogeneous boundary condition α ∂ u ∂ ν ( t , x ) + β u ( t , x ) ≥ w ( x ) on ( 0 , ∞ ) × ∂ Ω , $$\begin{array}{} \displaystyle \alpha \frac{\partial u}{\partial \nu}(t,x)+\beta u(t,x)\geq w(x)\quad\mbox{on } (0,\infty)\times \partial{\it\Omega}, \e…
Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds
2022
We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.