On some regularity results of parabolic problems with nonlinear perturbed terms and general data

In this paper, we study a class of nonlinear parabolic problems whose simplest model is given by $$\begin{aligned}({\mathcal {P}}_{1}) {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ a(t,x,u)(1+|u|)^{m}|\nabla u|^{p-2}\nabla u\right] =\mu \text { in }Q:=(0,T)\times \Omega ,\\ u(t,x)=0\text { on }(0,T)\times \partial \Omega ,\quad u(0,x)=u_{0}(x)\text { in }\Omega , \end{array}\right. }\end{aligned}$$ ( P 1 ) u t - div a ( t , x , u ) ( 1 + | u | ) m | ∇ u | p - 2 ∇ u = μ in Q : = ( 0 , T ) × Ω , u ( t , x ) = 0 on ( 0 , T ) × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in Ω , where $$\Omega $$ Ω is a bounded open set in $${\mathbb {R}}^{N}$$ R N ( $$N\ge 2$$ N ≥ 2 ), $$T>0$$ T > 0 , the vector filed $$\alpha \le a(t,x,u)\le \beta $$ α ≤ a ( t , x , u ) ≤ β (for some constants $$\alpha ,\beta >0$$ α , β > 0 ), $$m\ge 0$$ m ≥ 0 , $$u_{0}\in L^{1}(\Omega )$$ u 0 ∈ L 1 ( Ω ) and $$\mu $$ μ is a bounded Radon measure on Q. We show, under which condition on m, problem $$({\mathcal {P}}_{1})$$ ( P 1 ) admits a solution and we prove some regularity results. Moreover, in presence of a lower order (perturbed) term with natural growth $$\begin{aligned} ({\mathcal {P}}_{2}){\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ a(t,x,u)(1+|u|)^{m}|\nabla u|^{p-2}\nabla u\right] +(1+|u|)^{r-1}u|\nabla u|^{p}=\mu \\ \quad \text { in }Q:=(0,T)\times \Omega ,\\ u(t,x)=0\text { on }(0,T)\times \partial \Omega ,\quad u(0,x)=u_{0}(x)\text { in }\Omega , \end{array}\right. }\end{aligned}$$ ( P 2 ) u t - div a ( t , x , u ) ( 1 + | u | ) m | ∇ u | p - 2 ∇ u + ( 1 + | u | ) r - 1 u | ∇ u | p = μ in Q : = ( 0 , T ) × Ω , u ( t , x ) = 0 on ( 0 , T ) × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in Ω , we prove some a priori estimates on weak solutions for $$m>r+1$$ m > r + 1 ( $$r\in {\mathbb {R}}$$ r ∈ R ). Our methods rely on compactness arguments and convergence results, which give evidence of the optimality of the results.