Periodic solutions for doubly nonlinear evolution equations (Q639499)

The periodic problem \[ \begin{aligned} & A\Bigl(u'(t)\Bigr)+\partial \phi(u(t))\ni f(t)\;\text{ in}\;{ H},\;0<t<T,\\ & u(0)=u(T), \end{aligned} \] is considered, where \(u:[0, T]\to { H}\) is a trajectory in the Hilbert space \({ H}\), \(A\) is a maximal monotone operator in \({ H}\), \(\partial\phi\) denotes the subdifferential of a proper, lower-semicontinuous and convex functional \(\phi:{ H}\to [0, \infty]\), \(f\in L^2(0, T; { H})\). The authors give conditions for the operator \(A\), the functional \(\phi\) and for \(f\) under which the problem has at least one solution and conditions for uniqueness. As an application is considered the problem \[ \begin{aligned} &\gamma(u_t)-\Delta_p u\ni g\quad {\text{ in}}\quad \Omega \times (0, T),\\ &u=0\quad {\text{ on}}\quad \partial\Omega\times (0, T),\\ &u(\cdot, 0)=u(\cdot, T)\quad {\text{ in}}\quad \Omega, \end{aligned} \] where \(\gamma\) is a maximal monotone graph in \(\mathbb R^2\), \(g=g(x, t)\) is a given function, \(\Delta_p\) is the \(p\)-Laplace operator \(\Delta_p u(x)=\nabla\cdot\Bigl(|\nabla u(x)|^{p-2}\nabla u(x)\Bigr)\), \(1<p<\infty\), \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial\Omega\). The authors propose conditions for \(\gamma\) and \(g\) for which the problem admits at least one solution for \(p>{{2n}\over {n+2}}\) and conditions for uniqueness.