On the existence and uniqueness of solutions for a class of evolution equations (Q910915)

Let V and H be real separable Hilbert spaces such that \(V\subset H\subset V^*\) and \({\mathcal V}=L^ 2([0,T],V]\), \({\mathcal V}^*=L^ 2([0,T],V^*)\), \({\mathcal H}=L^ 2([0,T],H)\), \(Y={\mathcal V}\cap C([0,T],H)\). The following initial value problem is considered: \[ u'+A(u,u)+B(u,u)=0;\quad u(0)=a\in H,\quad u\in W=\{u\in {\mathcal V}:\quad u'\in {\mathcal V}^*\}, \] where A and B are nonlinear operators with \(A\in ({\mathcal H}\times {\mathcal V}\to {\mathcal V}^*)\) and \(B\in (Y\times Y\to {\mathcal V}^*)\). Suppose that A(\(\cdot,v)\) is continuous, A(u,.) is strongly monotonous and Lipschitz continuous and \[ <B(u,v),w>=\int^{T}_{0}b(u(t),v(t),w(t))dt, \] where \(b\in (V\times V\times V\to R)\) is 3-linear functional satisfying some conditions of growth. The authors prove the existence of a solution by using the fixed point theorem of Schauder. Under certain regularity conditions of operator A the problem has a unique solution.