Anti-periodic solutions to a class of non-monotone evolution equations. (English) Zbl 0961.34044

Aizicovici, Sergiu; Reich, Simeon

Discrete Contin. Dyn. Syst. 5, No. 1, 35-42 (1999).

The authors follow the ideas of H. Okochi on noncoercitive evolution equations with antiperiodic boundary conditions. The existence of solutions to the problem \[ au_t(t,x) +Au(t,x) -b u(t,x) +f(x, u(t,x))\ni h(t,x), \quad t\in[0,T], x\in\Omega, \]
\[ u(0,x)=-u(T,x), \] with \(h\in L^2(0,T; L_2(\Omega))\) is considered. Under the assumption that \(A\) is a maximal monotone operator in \(L^2(\Omega)\) and a subdifferential (in \(L^2(\Omega)\)) of a functional \(\varphi: L^2(\Omega)\to (-\infty, \infty], \;\) and some continuity assumption on the Carathéodory function \( f: \Omega\times {\mathbb{R}}\to {\mathbb{R}}\) it is proved that the problem has at least one generalized solution \(u\in W^{1,2}(0,T; L^2(\Omega))\). A combination of monotonicity and compactness methods is used. An example illustrating the abstract theory is discussed.

Reviewer: Irina V.Melnikova (Ekaterinburg)

MSC:

34G20	Nonlinear differential equations in abstract spaces
35R70	PDEs with multivalued right-hand sides
35K55	Nonlinear parabolic equations

Keywords:

anti-periodic solution; boundary value problem; subdifferential