Asymptotic behaviour of solutions to some pseudoparabolic equations (Q2785661)

The asymptotic behaviour of the solutions to the Cauchy problem for the following equation is investigated: NEWLINE\[NEWLINEu_t- \eta\Delta u_t-\nu\Delta u= f(x,u,\nabla u),\quad x\in\Omega\subset \mathbb{R}^n.\tag{1}NEWLINE\]NEWLINE In (1), \(\eta\) and \(\nu\) are nonnegative constants. Equation (1) arises in the theory of seepage of homogeneous fluids through a fissured rock, or also as the model of the unidirectional propagation of nonlinear, dispersive, long waves. In this paper, the case \(f(x,u,\nabla u)= \nabla\cdot F\), where \(F\in C^1(\mathbb{R},\mathbb{R}^n)\) is a fixed vector field, is studied. First, results of existence and regularity of solutions of (1) are given by using semigroup theory of linear operators, then the long time behaviour of solutions to the linearized equation is studied. Finally, the long time behaviour of solutions to the nonlinear Cauchy problem is examined.