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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.

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