zbMATH — the first resource for mathematics

On a class of nonlinear BBM-like equations. (English) Zbl 0914.35069
The paper is devoted to the Cauchy problem \[ \frac{du(t)}{dt} + A \frac{du(t)}{dt} = F(u(t)),\quad t>0,\quad u(0)=u_0, \tag{1} \] with a linear operator \(A\) such that \(-A\) generates an analytic semigroup in a Banach space \(X\) and a nonlinear operator \(F\) of the special form \(F=A^{\alpha}G\), \(G: X\to D(A^{\alpha})\), \(\alpha\in[0,1]\). Local weak solutions, that are solutions of the equation \[ u(t)=u_0+\int^t_0(I+A)^{-1}A^{\alpha}G(u(s))ds, \quad 0<t<T, \] and local (strong) solutions of (1) are under consideration. Well-posedness results are obtained. For the strong case they are the following:
Suppose \(G\) is a locally Lipschitz continuous mapping from \(X_1\) to \(X_{\alpha}\) (\(X_1, X_{\alpha}\) are \(D(A), D(A^{\alpha})\) with corresponding graph-norms). Then the Cauchy problem (1) has a unique solution for each initial value \(u_0\in X_1\).
An application to the Benjamin-Bona-Mahony (BBM) equation \[ \frac{\partial u(t,x)}{\partial t} - \Delta \frac{\partial u(t,x)}{\partial t} = \text{div } f(u(t,x)),\quad (t,x)\in (0,T)\times \Omega, \;\Omega\subset \mathbb{R}^n \] with the initial condition \(u(0,x)=u_0(x)\) and zero Dirichlet boundary condition, is given.

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
34G20 Nonlinear differential equations in abstract spaces
47H20 Semigroups of nonlinear operators