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Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data

Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data Abstract We study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂ ⁡ b ⁢ ( x , u ) ∂ ⁡ t - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) + g ⁢ ( x , t , u , ∇ ⁡ u ) + H ⁢ ( x , t , ∇ ⁡ u ) = μ in ⁢ Ω × ( 0 , T ) , ${\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),$ where the right-hand side belongs to L 1 ⁢ ( Q T ) + L p ′ ⁢ ( 0 , T ; W - 1 , p ′ ⁢ ( Ω ) ) ${L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}$ and b ⁢ ( x , u ) ${b(x,u)}$ is unbounded function of u , - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) ${{-}\operatorname{div}(a(x,t,u,\nabla u))}$ is a Leray–Lions type operator with growth | ∇ ⁡ u | p - 1 ${|\nabla u|^{p-1}}$ in ∇ ⁡ u ${\nabla u}$ . The critical growth condition on g is with respect to ∇ ⁡ u ${\nabla u}$ and there is no growth condition with respect to u , while the function H ⁢ ( x , t , ∇ ⁡ u ) ${H(x,t,\nabla u)}$ grows as | ∇ ⁡ u | p - 1 ${|\nabla u|^{p-1}}$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data

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Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
1072-947X
eISSN
1572-9176
DOI
10.1515/gmj-2016-0011
Publisher site
See Article on Publisher Site

Abstract

Abstract We study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂ ⁡ b ⁢ ( x , u ) ∂ ⁡ t - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) + g ⁢ ( x , t , u , ∇ ⁡ u ) + H ⁢ ( x , t , ∇ ⁡ u ) = μ in ⁢ Ω × ( 0 , T ) , ${\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),$ where the right-hand side belongs to L 1 ⁢ ( Q T ) + L p ′ ⁢ ( 0 , T ; W - 1 , p ′ ⁢ ( Ω ) ) ${L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}$ and b ⁢ ( x , u ) ${b(x,u)}$ is unbounded function of u , - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) ${{-}\operatorname{div}(a(x,t,u,\nabla u))}$ is a Leray–Lions type operator with growth | ∇ ⁡ u | p - 1 ${|\nabla u|^{p-1}}$ in ∇ ⁡ u ${\nabla u}$ . The critical growth condition on g is with respect to ∇ ⁡ u ${\nabla u}$ and there is no growth condition with respect to u , while the function H ⁢ ( x , t , ∇ ⁡ u ) ${H(x,t,\nabla u)}$ grows as | ∇ ⁡ u | p - 1 ${|\nabla u|^{p-1}}$ .

Journal

Georgian Mathematical Journalde Gruyter

Published: Sep 1, 2016

References