Double-phase parabolic equations with variable growth and nonlinear sources
نویسندگان
چکیده
Abstract We study the homogeneous Dirichlet problem for parabolic equations u t ? div ( class="MJX-tex-caligraphic" mathvariant="script">A z , ? ? ) = F x ? mathvariant="normal">? × 0 T {u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T), with double phase flux xmlns:m="http://www.w3.org/1998/Math/MathML"> ( width="-0.25em" p 2 + a q stretchy="false">) {\mathcal{A}}\left(z,| u=(| u{| }^{p\left(z)-2}+a\left(z)| }^{q\left(z)-2})\nabla u and nonlinear source F . The initial function belongs to a Musielak-Orlicz space defined by flux. functions , p q are Lipschitz-continuous, a\left(z) is nonnegative, may vanish on set of nonzero measure. exponents satisfy balance conditions N < ? r ? \frac{2N}{N+2}\lt {p}^{-}\le p\left(z)\le q\left(z)\lt p\left(z)+\frac{{r}^{\ast }}{2} {r}^{\ast }={r}^{\ast }\left({p}^{-},N) min Q stretchy="true">¯ width="0.33em" {p}^{-}={\min }_{{\overline{Q}}_{T}}\hspace{0.33em}p\left(z) It shown that under suitable growth F\left(z,u,\nabla u) respect second third arguments, has solution following properties: L ? 1 width="0.1em" for every s ? ; width="1em" mathvariant="normal">with max { } . \begin{array}{l}{u}_{t}\in {L}^{2}\left({Q}_{T}),\hspace{1.0em}| }^{p\left(z)+\delta }\in {L}^{1}\left({Q}_{T})\hspace{1.0em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}0\le \delta \lt },\\ | }^{s\left(z)},\hspace{0.33em}a\left(z)| }^{q\left(z)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega ))\hspace{1em}{\rm{with}}\hspace{0.33em}s\left(z)=\max \left\{2,p\left(z)\right\}.\end{array} Uniqueness proven stronger assumptions same results established regularized ? / {\mathcal{A}}(z,{({\varepsilon }^{2}+| }^{2})}^{1\text{/}2})\nabla > \varepsilon \gt 0
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ژورنال
عنوان ژورنال: Advances in Nonlinear Analysis
سال: 2022
ISSN: ['2191-950X', '2191-9496']
DOI: https://doi.org/10.1515/anona-2022-0271