On the Cauchy problem for evolution p(x)p(x)-Laplace equation

  • Stanislav Antontsev

    Universidade de Lisboa, Portugal
  • Sergey Shmarev

    Universidad de Oviedo, Spain


We consider the Cauchy problem for the equation

utdiv(a(x,t)up(x)2u)=f(x,t) in ST=Rn×(0,T)\text{$u_{t}-\operatorname{div} \left( a(x,t) |\nabla u|^{p(x)-2}\nabla u\right) =f(x,t)$ in $S_{T}=\mathbb{R}^{n}\times(0,T)$}

with measurable but possibly discontinuous variable exponent p(x):Rn[p,p+](1,)p(x):\,\mathbb{R}^{n}\mapsto [p^-,p^+]\subset (1,\infty). It is shown that for every u(x,0)L2(Rn)u(x,0)\in L^{2}(\mathbb{R}^{n}) and fL2(ST)f\in L^{2}(S_T) the problem has at least one weak solution uC0([0,T];Lloc2(Rn))L2(ST)u\in C^{0}([0,T];L^{2} _{loc}(\mathbb{R}^{n}))\cap L^{2}(S_{T}), up(x)L1(ST)|\nabla u|^{p(x)}\in L^{1}(S_{T} ). We derive sufficient conditions for global boundedness of weak solutions and show that the bounded weak solution is unique.

Cite this article

Stanislav Antontsev, Sergey Shmarev, On the Cauchy problem for evolution p(x)p(x)-Laplace equation. Port. Math. 72 (2015), no. 2/3, pp. 125–144

DOI 10.4171/PM/1961