Skip to main content

The local well-posedness of solutions for a nonlinear pseudo-parabolic equation

Abstract

The local existence and uniqueness of solutions for a nonlinear pseudo-parabolic equation are established in the Sobolev space C([0,T); H s ( R n )) C 1 ([0,T); H s 1 ( R n )) with s> n 2 . In addition, we prove the global existence of solutions for two special cases of the equation.

MSC: 35Q35, 35Q51.

1 Introduction

The pseudo-parabolic equation possesses the form

u t k u t =u+ u p ,x R n ,t>0,
(1)

where constant k>0, p>0, and = 1 n 2 x i 2 . If k=0, Eq. (1) becomes the heat equation with sources. If k>0, we call Eq. (1) as the pseudo-parabolic model (see Ting [1], Showalter and Ting [2]). The pseudo-parabolic equation has many important physical backgrounds such as the seepage of homogeneous fluids through a fissured rock [3], the unidirectional propagation of nonlinear dispersive long waves [4], [5] and the aggregation of populations [6] (where u is the population density). Equation (1) is employed in the analysis of nonstationary processes in the area of semiconductors [7], [8], where the term k u t u t is regarded as the free electron density rate, term u is regarded as the linear dissipation of the free charge current and u p is a source of free electron current. Equation (1) is also named a Sobolev type model or a Sobolev-Galpern type model [9].

The initial-boundary value problem and the initial problem for the linear pseudo-parabolic equation were investigated in [1], [2], [10] where the existence and uniqueness of solutions for the equation were established. Various dynamic properties of solutions for nonlinear pseudo-parabolic equations, including singular pseudo-parabolic equations and degenerate pseudo-parabolic equations can be found in [11]–[18]. It is worth to mention that Kaikina et al.[19] considered the superlinear case of the Cauchy problem for Eq. (1) with p>1 and showed the existence and uniqueness of the solutions. Furthermore, it was shown that the Cauchy problem for Eq. (1) has a unique global solution under the assumptions p>1+ 2 n and sufficiently small initial value u 0 . The existence, uniqueness, and comparison principle for mild solutions of Eq. (1) were established in Cao et al.[20] by whom the large time behavior of the solutions and the critical global existence exponent and the critical Fujita exponent for Eq. (1) were obtained.

In this work, we study the following nonlinear pseudo-parabolic equation:

u t k u t =u+α u q +βDf(u),x R n ,t>0,
(2)

where q1 is an integer, α and β are constants, f(u) is a polynomial with order m, f(0)=0, and D= 1 n x i . When β=0, Eq. (2) reduces to Eq. (1). The existence and uniqueness of local solutions for Eq. (2) are established in the Sobolev space C([0,T); H s ( R n )) C 1 ([0,T); H s 1 ( R n )) with s> n 2 . We find that the local solution in the space H s ( R n ) blows up if and only if lim t T u ( t , ) L ( R n ) =. For the space dimension n=1, assuming that the initial value u 0 H 1 ( R 1 ), α<0, and p is an odd number, we find the global existence of solutions for Eq. (2). For the other case n=1, p=1, and initial value u 0 H 1 (R), we also acquire the global existence result of solutions for Eq. (2).

The rest of this paper is organized as follows. The main results are stated in Section 2. Several lemmas and the proofs of main results are given in Section 3.

2 Main results

Firstly, we state some notations.

Let L p = L p ( R n ) (1p<+) be the space of all measurable functions h such that h L p p = R n | h ( t , x ) | p dx<. We define L = L ( R n ) with the standard norm h L = inf m ( e ) = 0 sup x R n e |h(t,x)|. For any real number s, H s = H s ( R n ) denotes the Sobolev space with the norm defined by

h H s = ( R n ( 1 + | ξ | 2 ) s | h ˆ ( t , ξ ) | 2 d ξ ) 1 2 <,

where h ˆ (t,ξ)= R n e i x ξ h(t,x)dx.

For T>0 and nonnegative number s, C([0,T); H s ( R n )) denotes the Frechet space of all continuous H s -valued functions on [0,T). We set Λ= ( 1 1 n 2 x i 2 ) 1 2 and Θ= ( 1 k ) 1 2 . For simplicity, throughout this article, we let c denote any positive constant.

We consider the Cauchy problem for Eq. (2)

{ u t k u t = u + α u q + β D f ( u ) , x R n , t > 0 , u ( 0 , x ) = u 0 ( x ) , x R n ,
(3)

which is equivalent to

{ u t = 1 k u + Θ 2 [ u k + α u q + β D f ( u ) ] , x R n , t > 0 , u ( 0 , x ) = u 0 ( x ) , x R n ,
(4)

where Θ 2 is the inverse operator of Θ 2 =1k.

Now, we give our main results for problem (3).

Theorem 2.1

Let u 0 (x) H s ( R n )withs> n 2 . Then the Cauchy problem (3) has a unique solutionu(t,x)C([0,T); H s ( R n )) C 1 ([0,T); H s 1 ( R n ))whereTis the maximum existence time. Moreover,

lim t T u ( t , ) H s ( R n ) =

if and only if

lim t T u ( t , ) L ( R n ) =.

For the case of space dimension n=1, we have the result.

Theorem 2.2

Letn=1, u 0 H 1 (R)in system (3), and assume thatqis an odd number andα0. Then problem (3) has a unique global solutionu(t,x)satisfying

u(t,x)C ( [ 0 , ) ; H s ( R ) ) C 1 ( [ 0 , ) ; H s 1 ( R ) ) ,s> 1 2 .

Theorem 2.3

Letn=1, q=1, and u 0 H 1 (R)in system (3). For any constantsαandβ, then problem (3) has a unique global solutionu(t,x)satisfying

u(t,x)C ( [ 0 , ) ; H s ( R ) ) C 1 ( [ 0 , ) ; H s 1 ( R ) ) ,s> 1 2 .

3 Several lemmas

Lemma 3.1

Letrandρbe real numbers such thatr<ρr. Then

u v H ρ ( R n ) c u H r ( R n ) v H ρ ( R n ) , if  r > n 2 , u v H r + ρ 1 / 2 ( R n ) c u H r ( R n ) v H ρ ( R n ) , if  r < n 2 .

This lemma can be found in [21] or [22].

Lemma 3.2

(Kato and Ponce [23])

Ifr0, then H r L is an algebra. Moreover,

u v H r ( R n ) c ( u L ( R n ) v H r ( R n ) + u H r ( R n ) v L ( R n ) ) ,

wherecis a constant depending only onr.

Lemma 3.3

Assume u 0 H s ( R n )withs> n 2 . Then problem (3) admits a unique local solution

u(t,x)C ( [ 0 , T ) ; H s ( R ) ) C 1 ( [ 0 , T ) ; H s 1 ( R ) ) .

Proof

For the first equation of problem (4), we have

u= u 0 + 0 t ( u k + Θ 2 [ u k + α u q + β D f ( u ) ] ) dt.
(5)

Letting functions u and v be in the closed ball B M 0 (0) of radius M 0 >1 about the zero function in C([0,T]; H s ( R n )) and letting Γ be the operator on the right-hand side of (5), for fixed t[0,T], we get

0 t ( u k + Θ 2 [ u k + α u q + β D f ( u ) ] ) d t 0 t ( v k + Θ 2 [ v k + α v q + β D f ( v ) ] ) d t H s T ( sup 0 t T u v H s ( R n ) + sup 0 t T u q v q H s ( R n ) + sup 0 t T f ( u ) f ( v ) H s ( R n ) ) .
(6)

Using Lemma 3.1 derives

u q v q H s ( R n ) = ( u v ) ( u q 1 + u q 2 v + + u v q 2 + v q 1 ) H s ( R n ) u v H s ( R n ) ( u q 1 + u q 2 v + + u v q 2 + v q 1 ) H s ( R n ) c M 0 q 1 u v H s ( R n )
(7)

and

f ( u ) f ( v ) H s ( R n ) c M 0 m 1 u v H s ( R n ) .
(8)

From (5)-(8), we obtain

Γ u Γ v H s θ u v H s ( R n ) ,
(9)

where θ=max(cT M 0 ,cT M 0 q 1 ,cT M 0 m 1 ) and c is independent of T. Choosing T sufficiently small such that θ<1, we know that operator Γ is a contractive mapping. Applying the above inequality and (5) yields

Γ u H s ( R n ) u 0 H s ( R n ) +θ u H s ( R n ) .
(10)

Choosing T sufficiently small such that θ M 0 + u 0 H s < M 0 , we know that Γ maps B M 0 (0) to itself. It follows from the contractive mapping principle that the mapping Γ has a unique fixed point u in B M 0 (0). This completes the proof. □

Lemma 3.4

Let functionu(t,x)be a solution of problem (3), s n 2 and the initial value u 0 (x) H s ( R n ). Forr(0,s1], there is a constantcdepending only on the coefficients of the first equation of system (3) such that

R ( Λ r + 1 u ) 2 d x R ( Λ r + 1 u 0 ) 2 d x + c 0 t ( 1 + u L ( R n ) q 1 + u L ( R n ) m 1 ) u H r + 1 ( R n ) 2 d τ .
(11)

Proof

Using Δ= Λ 2 +1 and the Parseval equality gives rise to

R Λ r u Λ r Δudx= R ( Λ r + 1 u ) Λ r + 1 udx+ R ( Λ r u ) 2 dx.

For r(0,s1], applying ( Λ r u) Λ r on both sides of the first equation of system (3), noting the above equality and integrating the resultant equation with respect to x by parts, we obtain the equation

1 2 d d t [ R ( ( Λ r u ) 2 + k ( Λ r u x ) 2 ) d x ] = R n ( Λ r + 1 u ) Λ r + 1 u d x + R n ( Λ r u ) 2 d x + α R n ( Λ r u ) Λ r ( u q ) d x + β R n ( Λ r u ) Λ r f ( u ) d x = I 1 + I 2 + I 3 + I 4 .
(12)

For the terms I 1 and I 2 , we have

| I 1 | u H r + 1 ( R n ) 2
(13)

and

| I 2 | u H r + 1 ( R n ) 2 .
(14)

For the terms I 3 and I 4 , using Lemma 3.2 gives rise to

| I 3 | Λ r u L 2 ( R n ) Λ r ( u q ) L 2 ( R n ) c u H r ( R n ) u L ( R n ) q 1 u H r ( R n ) c u L ( R n ) q 1 u H r + 1 ( R n ) 2
(15)

and

| I 4 | c Λ r u L 2 ( R n ) Λ r [ D f ( u ) ] L 2 ( R n ) c Λ r u L 2 ( R n ) Λ r + 1 f ( u ) L 2 ( R n ) c u H r ( R n ) ( 1 + u L ( R n ) m 1 ) u H r + 1 ( R n ) c ( 1 + u L ( R n ) m 1 ) u H r + 1 ( R n ) 2 .
(16)

It follows from (12)-(16) that

1 2 R [ ( Λ r u ) 2 + k ( Λ r u x ) 2 ] d x 1 2 R [ ( Λ r u 0 ) 2 + k ( Λ r u 0 x ) 2 ] d x c 0 t ( 1 + u L ( R n ) q 1 + u L ( R n ) m 1 ) u H r + 1 2 d τ ,

which results in (11). □

Proof of Theorem 2.1

Using Lemma 3.4, for any s> n 2 , we have

u H s ( R n ) c u 0 H s ( R n ) e 0 t [ 1 + u L ( R n ) q 1 + u L ( R n ) m 1 ] dt.
(17)

For s> n 2 , the Sobolev imbedding theorem yields

u L ( R n ) c u H s ( R n ) .
(18)

Applying the inequalities (17), (18), and Lemma 3.3 completes the proof. □

Proof of Theorem 2.2

For the space dimension n=1, we write problem (3) in the form

{ u t k u t x x = u x x + α u q + β [ f ( u ) ] x , x R , t > 0 , u ( 0 , x ) = u 0 ( x ) , x R .
(19)

Using R u j u x dx=0 for any integer j and integration by parts, we have

1 2 R u 2 d x = R u u t d x = R u [ k u t x x + u x x + α u q + β [ f ( u ) ] x ] d x = R [ k u x u t x u x 2 + α u q + 1 ] d x ,
(20)

which results in

1 2 d d t R ( u 2 + k u x 2 ) dx+ R u x 2 dxα R u q + 1 dx=0,
(21)

from which we obtain

1 2 R ( u 2 + k u x 2 ) dx+ 0 t R [ u x 2 α u q + 1 ] dxdt= 1 2 R ( u 0 2 + k u 0 x 2 ) dx.
(22)

If q is an odd integer, α0, and u 0 H 1 (R), we get

u ( t , ) L ( R ) c u 0 H 1 ( R ) .
(23)

Using the conclusion of Theorem 2.1, we finish the proof of Theorem 2.2. □

Proof of Theorem 2.3

For n=1 and q=1, using (22) yields

1 2 R ( u 2 + k u x 2 ) dx+ 0 t R [ u x 2 α u 2 ] dxdt= 1 2 R ( u 0 2 + k u 0 x 2 ) dx.
(24)

Since

| R [ u x 2 α u 2 ] d x | ( 1 + | α | ) u H 1 ( R ) 2 ,
(25)

it follows from (24) and (25) that

u H 1 ( R ) 2 u 0 H 1 ( R ) 2 e ( 1 + | α | ) t ,
(26)

from which we obtain

u L ( R ) u 0 H 1 ( R ) e ( 1 + | α | ) t ,
(27)

which together with Theorem 2.1 completes the proof of Theorem 2.3. □

References

  1. Ting TW: Parabolic and pseudo-parabolic partial differential equations. J. Math. Soc. Jpn. 1969, 21: 440-453. 10.2969/jmsj/02130440

    Article  Google Scholar 

  2. Showalter RE, Ting TW: Pseudoparabolic partial differential equations. SIAM J. Math. Anal. 1970, 1: 1-26. 10.1137/0501001

    Article  MathSciNet  Google Scholar 

  3. Barenblat G, Zheltov I, Kochiva I: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 1960, 24(5):1286-1303. 10.1016/0021-8928(60)90107-6

    Article  Google Scholar 

  4. Benjamin TB, Bona JL, Mahony JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 1972, 272: 47-78. 10.1098/rsta.1972.0032

    Article  MathSciNet  Google Scholar 

  5. Ting TW: Certain non-steady flows of second-order fluids. Arch. Ration. Mech. Anal. 1963, 14: 1-26.

    Article  Google Scholar 

  6. Padron V: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation. Trans. Am. Math. Soc. 2004, 356: 2739-2756. 10.1090/S0002-9947-03-03340-3

    Article  MathSciNet  Google Scholar 

  7. Korpusov MO, Sveshnikov AG: Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics. Zh. Vychisl. Mat. Mat. Fiz. 2003, 43(12):1835-1869.

    MathSciNet  Google Scholar 

  8. Korpusov MO, Sveshnikov AG: Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources. Differ. Uravn. 2006, 42: 404-415.

    MathSciNet  Google Scholar 

  9. Sobolev SL: On a new problem of mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat. 1954, 18: 3-50.

    MathSciNet  Google Scholar 

  10. Gopala RVR, Ting TW: Solutions of pseudo-heat equations in the whole space. Arch. Ration. Mech. Anal. 1972, 49: 57-78. 10.1007/BF00281474

    Article  Google Scholar 

  11. Brill H: A semilinear Sobolev evolution equation in a Banach space. J. Differ. Equ. 1977, 24: 412-425. 10.1016/0022-0396(77)90009-2

    Article  MathSciNet  Google Scholar 

  12. David C, Jet W: Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures. J. Math. Anal. Appl. 1979, 69: 411-418. 10.1016/0022-247X(79)90152-5

    Article  MathSciNet  Google Scholar 

  13. Karch G: Asymptotic behaviour of solutions to some pseudoparabolic equations. Math. Methods Appl. Sci. 1977, 20: 271-289. 10.1002/(SICI)1099-1476(199702)20:3<271::AID-MMA859>3.0.CO;2-F

    Article  MathSciNet  Google Scholar 

  14. Karch G: Large-time behaviour of solutions to nonlinear wave equations: higher-order asymptotics. Math. Methods Appl. Sci. 1999, 22: 1671-1697. 10.1002/(SICI)1099-1476(199912)22:18<1671::AID-MMA98>3.0.CO;2-Q

    Article  MathSciNet  Google Scholar 

  15. Ruggieri M, Speciale MP: Approximate analysis of a nonlinear dissipative model. Acta Appl. Math. 2014.

    Google Scholar 

  16. Kwek KH, Qu CC: Alternative principle for pseudo-parabolic equations. Dyn. Syst. Appl. 1996, 5: 211-217.

    MathSciNet  Google Scholar 

  17. Levine HA:Some nonexistence and instability theorems for solutions of formally parabolic equations of the form P u t =Au+F(u). Arch. Ration. Mech. Anal. 1973, 51: 371-386.

    Article  Google Scholar 

  18. Ptashnyk M: Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities. Nonlinear Anal. 2007, 66: 2653-2675. 10.1016/j.na.2006.03.046

    Article  MathSciNet  Google Scholar 

  19. Kaikina EI, Naumkin PI, Shishmarev IA: The Cauchy problem for a Sobolev type equation with power like nonlinearity. Izv. Math. 2005, 69: 59-111. 10.1070/IM2005v069n01ABEH000521

    Article  MathSciNet  Google Scholar 

  20. Cao Y, Yin JX, Wang CP: Cauchy problems of semilinear pseudo-parabolic equations. J. Differ. Equ. 2009, 246: 4568-4590. 10.1016/j.jde.2009.03.021

    Article  MathSciNet  Google Scholar 

  21. Kato T: Quasi-linear equations of evolution with applications to partial differential equations. In Spectral Theory and Differential Equations. Springer, Berlin; 1975:25-70. 10.1007/BFb0067080

    Chapter  Google Scholar 

  22. Lai SY, Wu M: The local strong and weak solutions to a generalized Novikov equation. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-134

    Google Scholar 

  23. Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1998, 41: 891-907. 10.1002/cpa.3160410704

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shaoyong Lai.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lai, S., Yan, H. & Wang, Y. The local well-posedness of solutions for a nonlinear pseudo-parabolic equation. Bound Value Probl 2014, 177 (2014). https://doi.org/10.1186/s13661-014-0177-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-014-0177-4

Keywords