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Double phase parabolic problem with variable growth | | Rakesh Arora
; Sergey Shmarev
; | | Date: |
16 Oct 2020 | | Abstract: | This paper addresses the questions of existence and uniqueness of strong
solutions to the homogeneous Dirichlet problem for the double phase equation
with operators of variable growth:
[ u_t - div left(|
abla u|^{p(z)-2}
abla u+ a(z) |
abla u|^{q(z)-2}
abla u
ight) = F(z,u) quad ext{in $Q_T=Omega imes (0,T)$} ] where
$Omega subset mathbb{R}^N$, $N geq 2$, is a bounded domain with the
boundary $partialOmegain C^2$, $z=(x,t)in Q_T$, $a:ar Q_T mapsto
mathbb{R}$ is a given nonnegative coefficient, and the nonlinear source term
has the form [ F(z,v)=f_0(z)+b(z)|v|^{sigma(z)-2}v. ] The variable exponents
$p$, $q$, $sigma$ are given functions defined on $ar{Q}_T$, $p$, $q$ are
Lipschitz-continuous and [ dfrac{2N}{N+2}<p^-leq p(z) leq q(z) < p(z) +
{frac{r}{2}} ext{with $0<r<r^ast=frac{4p^-}{2N + p^-(N+2)}$,quad
$p^-=min_{ar{Q}_T}p(z)$}. ] We find conditions on the functions $f_0$, $a$,
$b$, $sigma$ and $ u_0$ sufficient for the existence of a unique strong
solution with the following global regularity and integrability properties: [
egin{split} u_t in L^{2}(Q_T),quad & ext{$|
abla u|^{s(z)} in
L^{infty}(0,T;L^1(Omega))$ with $ s(z)=max{2,p(z)}$}, & |
abla
u|^{p(z)+delta}in L^1(Q_T)quad ext{for every $0<delta< r^*$}. {split} ]
The same results are established for the equation with the regularized flux
[ (epsilon^2+|
abla u|^2)^{frac{p(z)-2}{2}}
abla u + a(z)
(epsilon^2+|
abla u|^2)^{frac{q(z)-2}{2}}
abla u, qquad epsilon>0. ] | | Source: | arXiv, 2010.08306 | | Services: | Forum | Review | PDF | Favorites | |
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