Regularity for solutions of nonlinear evolution equations with nonlinear perturbations (Q1609058)

Let \(H\) and \(V\) be real separable Hilbert spaces such that \(V\) is a dense subset of \(H\). The following nonlinear functional-differential equation on \(H\) is considered \[ \frac{dx(t)}{dt}+Ax(t)=f\bigl(t,x(t)\bigr)+h(t), \quad 0< t\leq T,\qquad x(0)=x_0, \tag{1} \] where \(A: V\to V^*\) is a monotone and hemicontinuous operator from \(V\) to \(V^*\), where \(V^*\) denotes here the dual space of \(V\), such that \(A(0)=0\), \((Au-Av,u-v)\geq \omega_1\|u-v\|^2-\omega_2|u-v|^2\) and \(\|Au\|_*\leq\omega_3(\|u\|+1)\) for every \(u,v\in V\), where \(\omega_2\in \mathbb{R}\), \(\omega_1,\omega_3>0\) and \(\|\cdot\|\), \(|\cdot|\), \(\|\cdot\|_*\) denote the norms on \(V\), \(H\) and \(H^*\), respectively. Moreover, assume that \(f: [0,T]\times V\to H\) satisfies a Lipschitz condition with respect to the second variable. The main result of this paper states that, under the above assumptions for every \(h\in L^2(0,T;V^*)\) and \(x_0\in H\), equation (1) has a unique solution \[ x\in L^2(0,T;V)\cap C([0,T];H)\cap W^{1,2}(0,T;V^*) \] such that \[ \|x\|_{L^2\cap C\cap W^{1,2}} \leq c_2(1+|x|_0+\|h\|_{L^2(0,T;V^*)}), \] where \(c_2\) is a constant depending on \(T\).