On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations
Introduction
The paper is devoted to the problem of global existence and uniqueness of a classical solution to the following initial boundary value problem for the fourth order nonlinear Sobolev type equation:where b > −1, a > 0 and T > 0 are given numbers, f, g, h and ϕ are given functions and u is the unknown function.
We call a functions u(t,x) a classical solution of the Problem (1.1), (1.2), (1.3) if this function and all its derivatives involved in the Eq. (1.1) are continuous in QT ≔ [0, T] × [0, π] and the conditions (1.1), (1.2), (1.3) are satisfied in the usual sense.
There have been many works devoted to the study of initial boundary value problems for nonlinear Sobolev equations (see [3], [5], [10], [11] and references therein), where the problem of existence and uniqueness in appropriate Sobolev spaces, the problem of blow up of solutions and the problems of asymptotic behavior of solutions are studied.
As well as we know the equations considered in previous publications do not cover the class of equations we study.
The purpose of this paper is to prove existence and uniqueness of a global classical solution of the initial boundary value problem for the class of equations which covers the well-known viscous Cahn–Hilliard equation and the viscous Camassa–Holm equation. These equations appear in modeling dynamical processes in phase transitions and in modeling dynamics of shallow water waves (see [4], [6]).
In what follows we are using the following notations:
- •
By (·,·) and ∥·∥ we denote the inner product and norm in L2(0, π).
- •
We denote by the set of all functions u(t,x) of the form (2.1) such thatwhere α ⩾ 0, β ⩾ 1.
It is not difficult to see that this class of functions equipped with the normis a Banach space (see [8]).
We will use the inequalitythe well-known Wirtinger inequality (see [7])which are valid for each function such that w(0) = w(π) = 0 and the inequalitywhich is valid for each function such that w(0) = w(π) = 0.
The inequality (1.5) can be obtained easily from the equalityby using the Cauchy–Schwarz inequality. The inequality (1.7) follows from the relationand the inequality (1.6). Really by using the Cauchy–Schwarz inequality and the inequality (1.6) we obtain:HenceThis inequality and (1.6) imply (1.7).
We will employ also the following inequalitywhich is valid for each function ϕ that satisfies conditions:with a positive integer k.
Section snippets
Main result
Since the system of functions {sin(nx)}, n = 1, 2, … forms a basis in L2(0, π) the classical solution of Problem (1.1), (1.2), (1.3) has the formwhereBy using the Fourier’s method we can easily see that un(t), n = 1, 2,…, satisfy the following system of countably many nonlinear equations:where
Viscous Cahn–Hilliard equation
Let us note that whenthe Eq. (1.1) becomes the well-known viscous Cahn–Hilliard equationFor this equationIt is easy to check that all conditions of Theorem 2.9 are satisfied. Therefore, the initial boundary value problem for the equation under the initial condition (1.2) and the boundary conditions (1.3) has a unique global classical solution.
Viscous Camassa–Holm equation
Whenthe Eq.
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