*Accepted Paper*

**Inserted:** 20 apr 2021

**Last Updated:** 20 apr 2021

**Journal:** Memoris of the AMS

**Pages:** 126

**Year:** 2021

**Abstract:**

In this memoir, we study the existence and regularity of the quasilinear parabolic equations:

$$u_{t}-div(A(x,t,\nabla u))=B(u,\nabla u)+\mu,$$

in either $\mathbb{R}^{N+1}$ or $\mathbb{R}^N\times(0,\infty)$ or on a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$ where $N\geq 2$. In this paper, we shall assume that the nonlinearity $A$ fulfills standard growth conditions, the function $B$ is a continuous and $\mu$ is a radon measure.
Our first task is to establish the existence results with $B(u,\nabla u)=

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 $q>1$.

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