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Local behavior of solutions of quasilinear parabolic equations on metric spaces

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 Added by Janna Lierl
 Publication date 2017
  fields
and research's language is English
 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.
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