On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations

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Abstract

We prove the global existence and uniqueness of a classical solution to initial boundary value problem for a class of Sobolev type equations under the Dirichlet boundary conditions. This class of evolution equations covers the well-known viscous Cahn–Hilliard equation and the viscous Camassa–Holm equation.

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:-utxx+auxxxx+but=f(u)ux+(g(u,ux))x+h(u,ux,uxx,uxxx),t[0,T],x[0,π],u(0,x)=ϕ(x),x[0,π],u(t,0)=u(t,π)=uxx(t,0)=uxx(t,π)=0,t[0,T],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 Bα,Tβ the set of all functions u(t,x) of the form (2.1) such thatun(t)C[0,T]andn=1nαmax0tT|un(t)|β<,where α  0, β  1.

It is not difficult to see that this class of functions equipped with the normuα,T=n=1nαmax0tTn(t)|β1βis a Banach space (see [8]).

We will use the inequalitymaxx[0,π]|w(x)|π0π|w(x)|2dx1/2,the well-known Wirtinger inequality (see [7])0πw2(x)dx0π|w(x)|2dx,which are valid for each function wC1([0,π]) such that w(0) = w(π) = 0 and the inequality0πw2(x)dx0π|w(x)|2dx,which is valid for each function wC2([0,π]) such that w(0) = w(π) = 0.

The inequality (1.5) can be obtained easily from the equalityw(x)=0xw(y)dyby using the Cauchy–Schwarz inequality. The inequality (1.7) follows from the relation0=0π(w(x)w(x))dx=0π|w(x)|2+0πw(x)w(x)dxand the inequality (1.6). Really by using the Cauchy–Schwarz inequality and the inequality (1.6) we obtain:0π|w(x)|2dx0π|w(x)w(x)|dx0π|w(x)|21/20π|w(x)|21/20π|w(x)|21/20π|w(x)|21/2.Hence0π|w(x)|2dx1/20π|w(x)|2dx1/2.This inequality and (1.6) imply (1.7).

We will employ also the following inequalityn=1n2kϕn22πϕ(k)2,which is valid for each function ϕ that satisfies conditions:ϕC(k-1)[0,π],ϕ(k)L2(0,π),ϕ(2s)(0)=ϕ(2s)(π)=0,s=0,1,,k-12,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 formu(t,x)=n=1un(t)sin(nx),whereun(t)=2π0πu(t,x)sin(nx)dx,n=1,2,,t[0,T].By using the Fourier’s method we can easily see that un(t), n = 1, 2,…, satisfy the following system of countably many nonlinear equations:un(t)=ϕne-λnt+2π(b+n2)0t0πF(u(τ,x))e-λn(t-τ)sin(nx)dxdτ,n=1,2,,t[0,T],whereλn=an4b+n2,ϕn=2π0πϕ(x)sin(nx)dx,F(u)F(u,ux,uxx,uxxx)=f(u)ux+(g(u,ux))x+

Viscous Cahn–Hilliard equation

Let us note that whenf(u)0,h(u1,,u4)0,g(u1,u2)=λu2+3u12u2,λ>0,the Eq. (1.1) becomes the well-known viscous Cahn–Hilliard equation-utxx+auxxxx+ϵut=(au+bu3)xx.For this equationF(u)=(g(u,ux)x=(λux+3u2ux)x.It 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

Whenf(u)=-3u,g(u1,u2)0,h(u1,,u4)=u1u4+2u2u3the Eq.

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