The large-time asymptotic behavior of solutions to the periodic boundary value problem for a nonlinear Sobolev-type equation is studied. In particular, the case where the initial perturbations are not small is considered. In this case, the large-time behavior of solutions is dichotomous.
The aim of the present paper is to consider the Cauchy problem for the following Sobolev type equation with a power-law nonlinearity:where α > 0 and β = 0 are real numbers, and construct the asymptotics of the solution of problem (1) for large times t. Equation (1) can be viewed as a multidimensional generalization of the Boussinesq equation; it arises in liquid filtration problems for a porous medium with cracks [1, 2] and in the theory of crystalline semiconductors [3]. Suppose that the Fourier transformû 0 (p) of the initial condition u 0 (x) ∈ L 1 (R N ) satisfies the conditionswhere ε > 0 is sufficiently small.
Theorem 1. If condition (2) is satisfied, then the solution u(x, t) of the Cauchy problem (1) admits the asymptotic representationas t → ∞, and this asymptotics is uniform with respect to xt −1/2 ∈ R N , where the constant A 0 is given byLet us outline the proof of the theorem. (For details, see [4,5].) We apply the Fourier transform with respect to the variable x to the Cauchy problem (1); after the integration of the resulting relation with respect to t, we obtainwhere K(p) = α|p| 2 / (1 + |p| 2 ).By treating (5) as a functional equationû = A (û) and by using [6], we find that if the initial condition satisfies the condition u 0 (x) ∈ H (R N ), > N/2 + 2, and the norm u 0 H (RN ) is
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