Blow-up of solutions and local solvability of an abstract Cauchy problem of second order with a noncoercive source (Q6159048)

The equation at play is \[ \frac{d^2}{dt^2} \bigg( A_0u + \sum_{j=1}^n A_j(u)\bigg) + \frac{d}{dt}DP(u) + Lu = \frac{d}{dt} F(u), \quad u(0) = u_0 \quad u'(0) = u_1. \tag{1} \] The setup for the operator coefficients is: there are reflexive and separable Banach spaces \(V_0,V_1, \dots, V_n,\) \(W_1, W_2, W_3\) such that \begin{align*} A_0 &: V_0 \to V^*_0, \qquad \ A_j : V_j \to V^*_j, \qquad L : W_1 \to W_1^*, \\ F &: W_2 \to W_2^*, \qquad P : V_0 \to W_3, \qquad D : W_3 \to V^*_0 . \end{align*} There are dense and continuous imbeddings among the spaces, which put all terms of (1) in the same space. Among the assumptions: \(A_0\) is linear, continuous, symmetric and coercive. The \(A_j\) are continuous, monotonic, positive homogeneous and Fréchet differentiable. \(F\) is Lipschitz continuous on compact sets, positive homogeneous and Fréchet differentiable. \(D\) is linear and continuous. \(P\) is Lipschitz continuous on compact sets and Fréchet differentiable. \(L\) is linear, continuous, symmetric and coercive. Under these and various other conditions on the spaces, operators and Fréchet derivatives the authors give sufficient conditions for local existence of solutions of (1) and for solutions to blow up in finite time, with estimations for the blowup time.