Blow up of solutions for a class of fourth order nonlinear pseudo-parabolic equation with a nonlocal source

Abstract

In this paper, we consider the initial boundary value problem for a fourth order nonlinear pseudo-parabolic equation with a nonlocal source. By using the concavity method, we establish a blow-up result of the solutions under suitable assumptions on the initial energy.

Di and Shang Boundary Value Problems (2015) 2015:109 DOI 10.1186/s13661-015-0369-6

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Web End = Blow up of solutions for a class of fourth order nonlinear pseudo-parabolic equation with a nonlocal source

Huafei Di1,2 and Yadong Shang1,2*

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education, Guangzhou, 510006, China

Abstract

In this paper, we consider the initial boundary value problem for a fourth order nonlinear pseudo-parabolic equation with a nonlocal source. By using the concavity method, we establish a blow-up result of the solutions under suitable assumptions on the initial energy.

MSC: 35B44; 35K30; 35K59

Keywords: blow up; fourth order; nonlinear pseudo-parabolic; nonlocal source; concavity method

1 Introduction

In this article, we are concerned with the following initial boundary value problem:

ut u ut + u = up(x, t) K(x, y)up+(y, t) dy, x , t > ,

u(x, ) = u(x), x ,u = u = or u = u = , x , t > ,

(.)

where p > , and is a bounded domain of Rn (n ) with a smooth boundary . Here,

is the unit outward normal to , and K(x, y) is an integrable, real valued function such that K(x, y) = K(y, x). It is well known that this type of equations describes a variety of important physical processes, such as the analysis of heat conduction in materials with memory, viscous ow in materials with memory [], the theory of heat and mass exchange in stably stratied turbulent shear ow [], the non-equilibrium water-oil displacement in porous strata [], the aggregation of populations [], the velocity evolution of ion-acoustic waves in a collisionless plasma when ion viscosity is invoked [], ltration theory [, ], cell growth theory [, ], and so on. In population dynamics theory, the nonlocal term indicates that evolution of species at a point of space does not depend only on the nearby density but also on the total amount of species due to the eects of spatial inhomogeneity; see [].

There have also been many profound results on the existence of global solutions and asymptotic behavior of the solutions for the initial boundary value problems and the initial value problems of fourth order nonlinear pseudo-parabolic equations.

2015 Di and Shang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License

(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro

vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and

indicate if changes were made.

Di and Shang Boundary Value Problems (2015) 2015:109 Page 2 of 9

In , Kabanin [] considered the following problem:

R, t > ,

where > , , > are constants. They showed that the solutions of this problem

are expressed through the sum of convolutions of functions (x) and f (t, x) with corresponding fundamental solutions of the problem.

Zhao and Xuan [] studied the following fourth order pseudo-parabolic equation:

ut uxx uxxt + uxxxx + f (u)x = , x

R, t . (.)

They obtained the existence of the global smooth solutions for the initial value problem of (.) and discussed the convergence behavior of solutions as .

Recently, Khudaverdiyev and Farhadova [] discussed the following fourth order semi-linear pseudo-parabolic equation:

ut uxxt + uxxxx = f (t, x, u, ux, uxx, uxxx), x , t T, (.)

with Ionkin type non-self-adjoint mixed boundary conditions, where > is a xed number. They proved the local existence for a generalized solution of the mixed problem under consideration by combining generalized contracted mapping principle and Schauders xed point principle and then proved the global existence for a generalized solution by means of Schauders stronger xed point principle.

The so-called viscous Cahn-Hilliard equation is also in a class of fourth order nonlinear pseudo-parabolic equations and can be considered as a special case of (.). In recent years, a lot of attention has been paid to the viscous Cahn-Hilliard equations. For more and deeper investigations of the stability analysis (as t ) and the asymptotic behavior of

viscous Cahn-Hilliard models, we refer readers to [, ] and the references therein.

Since the study on blow-up solutions for nonlinear parabolic equation with nonlocal source by Levine in [], many eorts have been made devoted to the study of blow-up properties for nonlocal semilinear parabolic equations. The upper bound and lower bound of the blow-up time, blow-up rate estimate, blow-up set, and blow-up prole of the blowup solutions for a various of nonlocal semilinear parabolic equations with nonlocal source terms or nonlocal boundary condition have been widely studied in the last few decades; we refer the readers to [] and the references cited therein.

Korpusov [] considered a Sobolev type equation with a nonlocal source and obtained blow-up results under suitable conditions on initial data and nonlinear function. In [],

ut uxxt + uxxxx = uxx, < x < l, t > , u(x, ) = (x), x l,u(, t) = u(l, t) = uxx(, t) = uxx(l, t) = , t T,

(.)

where , , are positive constants. A classical solution of this mixed problem is obtained through the Fourier method in the form of a series. Conditions sucient for uniform convergence of this series are found.

In , Bakiyevich and Shadrin [] considered the following problem:

ut uxxt + uxxxx = uxx + f (t, x), x

R,

(.)

u(x, ) = (x), x

Di and Shang Boundary Value Problems (2015) 2015:109 Page 3 of 9

Bouziani studied the solvability of nonlinear pseudo-parabolic equation with a nonlocal boundary condition. More results on the global well-posedness for the nonlinear pseudo-parabolic equation with nonlocal source can be found in [] and the references therein.

Motivated by the above-mentioned works, we investigate the blow-up behavior of solutions of the initial boundary value problem for a fourth order nonlinear pseudo-parabolic equation with a nonlocal source (.). By using the concavity method, we prove a nite time blow-up result under some assumption on the initial energy E().

2 Preliminaries

In this section, we rst state a local existence theorem, which can be obtained by Faedo-Galerkin methods. The interested readers are referred to Lions [] or Escobedo and Herrero [] for details.

Theorem . Assume that p > and u H( ). Then there exists a Tm > for which

problem (.) has a unique local solution u C([, Tm); H( )) satisfying

(ut, v) + (u, v) + (ut, v) + ( u, v) =

up(x, t)

K(x, y)up+(y, t) dy, v

, (.)

for all v H( ) and t [, Tm).

Before stating our principal theorem, we note that the Frchet derivative fu of the nonlinear function f (u) = up(x, t) K(x, y)up+(y, t) dy is

fu h(x, t) = pup(x, t)h(x, t)

K(x, y)up(y, t)h(y, t) dy, u H( ).

Clearly fu is symmetric and bounded, so that the potential F exists and is given by

F(u) =

f (u), u d

u(x, t) dx d

= pup(x, t)

K(x, y)p+up+(y, t) dy

K(x, y)up+(x, t)up+(y, t) dx dy. (.)

Now, dierentiating the identity (.) with respect to t, it follows that

ddt F(u) =

p +

=

p +

d dt

K(x, y)up+(x, t)up+(y, t) dx dy

=

K(x, y)up(x, t)up+(y, t)ut(x, t) dx dy +

K(x, y)up(y, t)up+(x, t)ut(y, t) dx dy =

K(x, y)up(x, t)up+(y, t)ut(x, t) dx dy =

f (u), ut , (.)

where we have used the symmetry of K(x, y).

Di and Shang Boundary Value Problems (2015) 2015:109 Page 4 of 9

To obtain the blow-up result, we will introduce the energy function. We have

E(t) =

| u| dx

|u| dx +

K(x, y)up+(x, t)up+(y, t) dx dy. (.)

Lemma . Let p > and u be a solution of the problem (.). Then E(t) is non-increasing function, that is, E (t) . Moreover, the following energy equality holds:

E(t) +

t

p +

|ut| + |ut| dx d = E().

Proof Multiplying (.) by ut and integrating over , we have

|ut| dx +

|ut| dx +

d dt

|u| dx +

d dt

| u| dx

ut(x, t) dx.

= up(x, t)

K(x, y)up+(y, t) dy

Hence, from (.), we obtain

|ut| dx +

|ut| dx +

d dt

|u| dx +

d dt

| u| dx

=

p +

d dt

K(x, y)up+(x, t)up+(y, t) dx dy

and

|ut| dx +

|ut| dx +

ddt E(t) = . (.)

Integrating (.) from to t, we nd

E(t) +

t

|ut| + |ut| dx d = E(). (.)

The proof of the Lemma . is completed.

3 Blow up of solutions

Now, we will state the blow-up result of the solutions to the problem (.).

Theorem . Assume that p > and u H( ). If u(x, t) is a solution of the problem (, ) and the initial data u(x) satises

|u| + |u| dx > E(), (.)

then the solution of problem (.) blows up in nite time; that is, the maximum existence time Tmax of u(x, t) is nite and

lim

tT

max

t

|u| + |u| dx d = +,

Di and Shang Boundary Value Problems (2015) 2015:109 Page 5 of 9

where = m; m = ( ); p + ; is the rst eigenvalue of operator under homogeneous Dirichlet boundary conditions.

Proof The proof makes use of the so-called concavity method. Multiplying (.) by u and integrating over , we have

d dt

|u| dx +

d dt

|u| dx +

| u| dx

|u| dx +

u(x, t) dx.

= up(x, t)

K(x, y)up+(y, t) dy

Hence

d dt

|u| dx +

d dt

|u| dx +

| u| dx

|u| dx +

K(x, y)up+(x, t)up+(y, t) dx dy + E(u)

|u| dx

| u| dx +

K(x, y)up+(x, t)up+(y, t) dx dy

=

d dt

|u| dx

+ E(u)

|u| dx +

+ p +

K(x, y)up+(x, t)up+(y, t) dx dy

+

|u| dx +

| u| dx = . (.)

We consider the following function:

H(t) =

|u| dx + |u| dx E(). (.)

From (.), (.), Lemma ., and Poincars inequality, we have

ddt H(t)

=

d dt

|u| + |u| dx

=

|u| + | u| dx E(u)

+ p +

K(x, y)up+(x, t)up+(y, t) dx dy

=

|u| + | u| dx E(u) +

t

|ut| + |ut| dx d

+ p +

K(x, y)up+(x, t)up+(y, t) dx dy

|u| + | u| dx E(u)

Di and Shang Boundary Value Problems (2015) 2015:109 Page 6 of 9

|u| + |u| dx E(u)

= m

|u| + |u| dx E(u)

= mH(t), (.)

where = m; m = ( ); p + ; is the rst eigenvalue of operator under homogeneous Dirichlet boundary conditions.

Due to the conditions (.), it follows that

H() =

|u| + |u| dx E(u) > . (.)

Multiplying (.) by emt, we have

emt ddt H(t) memtH(t) =

d dt

emtH(t) .

From the last inequality above and (.), we obtain

H(t) H()emt > . (.)

From what has been discussed above, we nd

d dt

|u| + |u| dx >

t

|ut| + |ut| dx d. (.)

Now we dene

G(t) =

t

|u| + |u| dx d. (.)

Dierentiating the identity (.) with respect to t, we deduce that

G (t) =

|u| + |u| dx,

G (t) = d dt

|u| + |u| dx

t

|ut| + |ut| dx d,

so we have

G (t)G(t)

t

|ut| + |ut| dx d

t

|u| + |u| dx d

t

|ut| dx d

t

|u| dx d

+

t

|ut| dx d

t

|u| dx d

+

t

|ut| dx d

t

|u| dx d

+

t

|ut| dx d

t

|u| dx d. (.)

Di and Shang Boundary Value Problems (2015) 2015:109 Page 7 of 9

Using Schwarzs inequality, we get

t

uut dx d

t

|ut| dx d

t

|u| dx d, (.)

t

uut dx d

t

t

|u| dx d, (.)

and

|ut| dx d

t

uut dx d

t

uut dx d

t

|ut| dx d

t

|u| dx d

t

t

|u| dx d

|ut| dx d

t

|ut| dx d

t

|u| dx d

t

t

|u| dx d. (.)

Inserting (.)-(.) into (.), we nd

G (t)G(t)

t

+ |ut| dx d

uut dx d

+

t

uut dx d

+

t

uut dx d

t

uut dx d

=

t

(uut + uut) dx d

t

= G (

) d

=

G (t) G () . (.)

Thus, we obtain

G (t)G(t)

G (t) G () . (.)

On the other hand, from (.), we know

lim

t H(t) = +.

This implies

G (t) =

|u| + |u| dx +, t . (.)

Hence, for < < there exists a T, such that for all t T

G (t) G () G (t). (.)

Di and Shang Boundary Value Problems (2015) 2015:109 Page 8 of 9

By (.) and (.), we have

G (t)G(t)

G (t)G(t) < , t T. (.)

Since a concave function must always lie below any tangent line, we see that G(t)q reaches in nite time as t T, where T > T. This means

lim

tT

G(t) = +,

t

|u| + |u| dx d = +. (.)

Then the desired assertion immediately follows.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed to each part of this work equally and read and approved the nal manuscript.

Acknowledgements

This work is supported by the NSF of China (11401122, 40890153), the Scientic Program (2008B080701042) of Guangdong Province.

Received: 12 March 2015 Accepted: 3 June 2015

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