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Title: On some nonlinear parabolic equations with nonmonotone multivalued terms
Author: Otani, Mitsuharu
Staicu, Vasile
Keywords: Inear parabolic equations
Nonmonotone multivalued terms
Initial-boundary value problems
Existence results
Local solutions
Small and large global solutions
Issue Date: Jan-2021
Publisher: Heldermann Verlag
Abstract: We study the local and the global existence of solutions to a class of nonlinear parabolic initial-boundary value problems driven by the equation \[ \frac{\partial u\left( x,t\right) }{\partial t}-\triangle u\left( x,t\right) \in-\partial\Phi\left( u\left( x,t\right) \right) +G\left( x,t,u\left( x,t\right) \right) ,\left( x,t\right) \in Q_{T}, \] where $\partial\Phi$ denotes the subdifferential (in the sense of convex analysis) of a proper, convex and lower semicontinuous function $\Phi \colon\mathbb{R\rightarrow}\left[0, \infty\right]$, $\Omega\subseteq \mathbb{R}^{N}$ is a bounded open set, $T>0,$ $Q_{T}:=\Omega\times\left[0,T\right]$, and $G \colon Q_{T}\times\mathbb{R\rightarrow}2^{\mathbb{R}}$ is a multivalued mapping whose growth order with respect to $u$ is Sobolev sub-critical. \par\vskip1mm We prove two local existence results: one for the case where the multivalued mapping $u\mapsto G\left(\cdot, \cdot,u\right)$ is upper semicontinuous with closed convex values and the second one deals with the case when $u\mapsto G\left(\cdot, \cdot, u\right)$ is lower semicontinuous with closed (not necessarily convex) values. We also give two types of results concerning the global continuation of local solutions. One is for any large data and the other one for small data. Finally, we exemplify the applicability of our results.
Peer review: yes
ISSN: 0944-6532
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Appears in Collections:CIDMA - Artigos
FAAG - Artigos

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