Variational analysis of the nerve equation
The author describes a variational method which may be used to find periodic solutions for a class of partial differential equations including the model of the nerve equation \[ {\partial x\over\partial t}= {\partial^2x\over\partial u^2}+ f,\tag{1} \] where \(f(t,x)\) is a nonlinear function, \(t\in\mathbb{R}\), \(u\in\Omega\subset\mathbb{R}\). The equation (1) is considered as a special case of the equation \({\partial x\over\partial t}= Bx+f(\cdot,x(\cdot))\), where \(B\) is a differential operator defined on a functional space. A family of continuous differential operators \(A(t): X\to X^*\) (\(X\) is a complex reflexive Banach space, \(X^*\) is its dual) is considered and, also, some suitable functionals on the space \(W_{\text{per}}(0,T)= \{x\in W(0,T): x(0)=x(T)\}\), in which \(W(0,T)= \{x\in L^2(0,T; X): \dot x\in L^2(0,T;X^*)\}\).
scientific article; zbMATH DE number 1014353