Potential Estimates and Quasilinear Parabolic Equations with Measure Data

Abstract

In this memoir, we study the existence and regularity of the quasilinear parabolic equations: u t − div ⁡ ( A ( x , t , ∇ u ) ) = B ( u , ∇ u ) + μ , \begin{equation*} u_t-\operatorname {div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu , \end{equation*} in either R N + 1 \mathbb {R}^{N+1} or R N × ( 0 , ∞ ) \mathbb {R}^N\times (0,\infty ) or on a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 \Omega \times (0,T)\subset \mathbb {R}^{N+1} where N ≥ 2 N\geq 2 . We shall assume that the nonlinearity A A fulfills standard growth conditions, the function B B is a continuous and μ \mu is a radon measure. Our first task is to establish the existence results with B ( u , ∇ u ) = ± | u | q − 1 u B(u,\nabla u)=\pm |u|^{q-1}u , for q > 1 q>1 . We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with B ≡ 0 B\equiv 0 , under minimal conditions on the boundary of domain and on nonlinearity A A . Finally, due to these estimates, we solve the existence problems with B ( u , ∇ u ) = | ∇ u | q B(u,\nabla u)=|\nabla u|^q for q > 1 q>1 .