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We consider the nonlinear hyperbolic‐type inequality under the Dirichlet‐type boundary condition where , and . An optimal Fujita‐type result is obtained for this problem. Namely, we prove that when belongs to a certain functional space , and or , then there exists no global weak solution, while global solutions exist for some when and . The obtained result shows an interesting phenomenon of discontinuity of the Fujita critical exponent, jumping from to as reaches the value from above.
We consider the nonlinear hyperbolic‐type inequality under the Dirichlet‐type boundary condition where , and . An optimal Fujita‐type result is obtained for this problem. Namely, we prove that when belongs to a certain functional space , and or , then there exists no global weak solution, while global solutions exist for some when and . The obtained result shows an interesting phenomenon of discontinuity of the Fujita critical exponent, jumping from to as reaches the value from above.
In this paper, we consider a peridynamic model with energy damping inspired by the works of Balakrishnan and Taylor on “damping models” based on the instantaneous total energy of the system. We study the asymptotic behavior of solutions, in the sense of attractors, of these peridynamic models in suitable phase space; more precisely, we prove a result of existence and characterization of compact global attractors with a nonlinear strongly continuous semigroup approach based in the asymptotic smoothness property thanks to Chueshov and Lasiecka and Nakao's lemma.
This research establishes the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator depending on three real parameters, originated from a capillary phenomena, of the following form: $$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle -\Delta ^{l}_{p(x)}u+\delta \vert u\vert ^{\alpha (x)-2}u=\mu g(x, u)+\lambda f(x, u, \nabla u) &{} \mathrm {i}\mathrm {n}\ \Omega ,\\ \\ u=0 &{} \mathrm {o}\mathrm {n}\ \partial \Omega , \end{array}\right. \end{aligned}$$ - Δ p ( x ) l u + δ | u | α ( x ) - 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) i n Ω , u = 0 o n ∂ Ω , where $$\Delta ^{l}_{p(x)}$$ Δ p ( x ) l is the p(x)-Laplacian-like operator, $$\Omega $$ Ω is a smooth bounded domain in $$\mathbb {R}^{N}$$ R N , $$\delta ,\mu $$ δ , μ , and $$\lambda $$ λ are three real parameters, and $$p(\cdot ),\alpha (\cdot )\in C_{+}(\overline{\Omega })$$ p ( · ) , α ( · ) ∈ C + ( Ω ¯ ) and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized $$(S_{+})$$ ( S + ) type and the theory of variable-exponent Sobolev spaces, we establish the existence of a weak solution for the above problem.
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