Nonlinear parabolic equations with \(L^1\) data as limit of bilateral problems (Q2774113)

The author considers the following parabolic problem NEWLINE\[NEWLINE \begin{cases} u_t-\text{div } a(x,t,Du)=f(x,t) \quad \text{in} Q,\\ u(x,0)=u_0(x) \quad \text{in} \Omega,\\ u(x,t)=0 \quad \text{on} \Gamma.\end{cases} NEWLINE\]NEWLINE The function \(a:Q\times {\mathbb R}^n\to {\mathbb R}^n,\) \(n\geq 2\) is a Carathéodory function satisfying certain conditions under which the operator NEWLINE\[NEWLINEA(u)=-\text{div}(a(x,t,Du)) : L^p(0,T; W_0^{1,p}(\Omega))\to L^{p'}(0,T; W_0^{-1,p'}(\Omega)), \quad {1\over p}+{1\over p'}=1NEWLINE\]NEWLINE is a coercive, continuous, bounded and strictly monotone one. It is obtained existence of entropy solution \(u\in C([0,T]; L^1(\Omega))\) of the above problem as a limit of the solution \(u^n\) as \(n\to \infty\) of the corresponding bilateral problem NEWLINE\[NEWLINE \int_0^T\langle u_t^n,v-u^n\rangle+\iint_Qa(x,t,Du^n) D(v-u^n)\geq \iint_Q f(v-u^n) NEWLINE\]NEWLINE with \(f\in L^1(Q),\) \(u_0\in L^1(\Omega)\) and \(v\in L^p(0,T;W^{1,p}_0(\Omega)),\) \(|v|\leq n\) a.e. in \(Q\).