POTENTIAL ESTIMATES AND QUASILINEAR PARABOLIC EQUATIONS WITH MEASURE DATA
Résumé
In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\text{div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu$$ in $\mathbb{R}^{N+1}$, $\mathbb{R}^N\times(0,\infty)$ and a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$. Here $N\geq 2$, the nonlinearity $A$ fulfills standard growth conditions and $B$ term is a continuous function and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm|u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=|\nabla u|^q$ for $q>1$.
Mots clés
- Riccati type parabolic equation.
- Lane-Emden type parabolic equation
- Reifenberg flat domain
- Uniformly thick domain
- Decay estimate
- Capacitary estimate
- Lorentz-Morrey space
- Lorentz space
- Parabolic Bessel kernel
- Heat kernel
- Fractional Maximal parabolic potential
- Riezs parabolic potential
- Quasilinear parabolic equations
- Riccati type parabolic equation
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