Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}\)-data
The authors prove well-posedness, existence and uniqueness of a renormalized solution of a nonlinear parabolic equation with variable exponents and \(L^1\)-data, namely \[ \begin{cases} \partial_t u-\text{div}(|\nabla u|^{p(x)-2}\nabla u)=f &\text{in}\;Q_T,\\ u=0 &\text{on}\;\Sigma_T,\\ u(0,\cdot)=u_0(\cdot) &\text{in}\;\Omega, \end{cases} \] where \(\Omega\subset \mathbb R^ N\) is a~bounded open Lipschitz domain, \(N\geq2\), \(Q_Y=(0,T)\times\Omega\) and \(\Sigma_T=(0,T)\times\partial\Omega\). The authors work in the functional setting of the Lebesgue and Sobolev variable-exponent spaces. The main result states that when the variable-exponent function \(p:\overline{\Omega}\to[1,\infty)\) is continuous and moreover satisfies \[ 1<\min_{x\in\overline{\Omega}} p(x)\leq\max_{x\in\overline{\Omega}} p(x)<N, \] then there exists at least one renormalized solution \(u\) to the above-stated problem. Moreover, if \(\min_{x\in\overline{\Omega}} p(x)>2-\frac1{N+1}\), then \(u\in L^{q^-}(0,T;W_0^{1,q(\cdot)}(\Omega))\) for every continuous variable-exponent function \(q\) satisfying \[ 1\leq q(x)<\frac{N(p(x)-1)+p(x)}{N+1} \text{ for all } x\in\overline{\Omega}, \text{ where }q^-=\min_{x\in\overline{\Omega}} q(x). \] The proof is deep and involved and several side results of independent interest are established in the course of it. The techniques used in the proof involve the semigroup approach, a rather comprehensive study of various properties of renormalized solutions, a priori estimates, various results on several types of convergence of functions approximating the solution with a special emphasis on strong convergence. The uniqueness of the solution is proved separately. Some connections to entropy solutions are observed.
scientific article; zbMATH DE number 5787090
scientific article