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Register a new userLocal behavior of solutions of quasilinear parabolic equations on metric spaces
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Authors
Janna Lierl
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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.
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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 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.
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.
We describe the behavior of p-harmonic Greens functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincare inequality.
We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations that involves both non-degenerate and singular operators. Throughout a parabolic approach to expansion of positivity we obtain the interior Holder continuity, and some integral and pointwise Harnack inequalities.
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