Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions
106
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{rm div}left(g(x,t,| abla u|),frac{ abla u}{| abla u|}right)=0, $$ under the generalized non-logarithmic Zhikovs condition $$ g(x,t,{rm v}/r)leqslant c(K),g(y,tau,{rm v}/r), quad (x,t), (y,tau)in Q_{r,r}(x_{0},t_{0}), quad 0<{rm v}leqslant Klambda(r), $$ $$ quad limlimits_{rrightarrow0}lambda(r)=0, quad limlimits_{rrightarrow0} frac{lambda(r)}{r}=+infty, quad int_{0} lambda(r),frac{dr}{r}=+infty. $$ In particular, our results cover new cases of double-phase parabolic equations.
rate research
Read More
We bound the difference between solutions $u$ and $v$ of $u_t = aDelta u+Div_x f+h$ and $v_t = bDelta v+Div_x g+k$ with initial data $phi$ and $ psi$, respectively, by $Vert u(t,cdot)-v(t,cdot)Vert_{L^p(E)}le A_E(t)Vert phi-psiVert_{L^infty(R^n)}^{2rho_p}+ B(t)(Vert a-bVert_{infty}+ Vert abla_xcdot f- abla_xcdot gVert_{infty}+ Vert f_u-g_uVert_{infty} + Vert h-kVert_{infty})^{rho_p} abs{E}^{eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $xinR^n$, and $t$. The functions $a$ and $h$ may in addition depend on $ abla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $EsubsetR^n$ is assumed to be a bounded set, and $rho_p$ and $eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
We introduce elliptic and parabolic $mathcal{B}_{1}$ classes that generalize the well-known $mathfrak{B}_{p}$ classes of DeGiorgi, Ladyzhenskaya and Uraltseva with $p>1$. New classes are applied to prove pointwise continuity of solutions of elliptic and parabolic equations with nonstandard growth conditions. Our considerations cover new cases of variable exponent and $(p, q)$-phase growth including the ,,singular-degenerate parabolic case $p<2<q$.
We prove continuity and Harnacks inequality for bounded solutions to elliptic equations of the type $$ begin{aligned} {rm div}big(| abla u|^{p-2}, abla u+a(x)| abla u|^{q-2}, abla ubig)=0,& quad a(x)geqslant0, |a(x)-a(y)|leqslant A|x-y|^{alpha}mu(|x-y|),& quad x eq y, {rm div}Big(| abla u|^{p-2}, abla u big[1+ln(1+b(x), | abla u|) big] Big)=0,& quad b(x)geqslant0, |b(x)-b(y)|leqslant B|x-y|,mu(|x-y|),& quad x eq y, end{aligned} $$ $$ begin{aligned} {rm div}Big(| abla u|^{p-2}, abla u+ c(x)| abla u|^{q-2}, abla u big[1+ln(1+| abla u|) big]^{beta} Big)=0,& quad c(x)geqslant0, , betageqslant0,phantom{=0=0} |c(x)-c(y)|leqslant C|x-y|^{q-p},mu(|x-y|),& quad x eq y, end{aligned} $$ under the precise choice of $mu$.
We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Omegatimes(0,+infty)$, where $Omega$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
