Volume 2010, Article ID 872572,6pages doi:10.1155/2010/872572
Research Article
Continuous Dependence for
the Pseudoparabolic Equation
M. Yaman and S¸. G ¨ur
Department of Mathematics, Sakarya University, 54100 Sakarya, Turkey
Correspondence should be addressed to M. Yaman,[email protected]
Received 29 April 2010; Revised 7 July 2010; Accepted 18 August 2010
Academic Editor: Gary Lieberman
Copyrightq2010 M. Yaman and S¸. G ¨ur. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We determine the continuous dependence of solution on the parameters in a Dirichlet-type initial-boundary value problem for the pseudoparabolic partial differential equation.
1. Introduction
We consider the following initial-boundary value problem:
ut−αΔut−βΔufu, x∈Ω, t >0, 1.1
ux,0 u0x, x∈Ω, 1.2
ux, t 0, x∈∂Ω, t >0, 1.3
whereαandβare positive constants,Ω⊂Rnis a bounded domain with sufficiently smooth
boundary∂Ω,andfuis a given nonlinear function which satisfies
0≥Fu≥fu·u, 1.4
fu≤c1
1 |u|p, 1.5
whereFu 0ufsds,c1is a positive constant, andp≤n/n−2.
A mixed-boundary value problem for the one-dimensional case of1.1appears in the study of nonsteady flow of second-order fluids1where u represents the velocity of the fluid.
Equation 1.1 can be assumed as a model for the heat conduction involving a thermodynamic temperatureθu−αΔuand a conductive temperatureu; see2.
Equations of the form1.1have been called pseudoparabolic by Showalter and Ting 3, because well posed initial-boundary value problems for parabolic equations are also well-posed for1.1. Moreover, in certain cases, the solution of a parabolic initial-boundary value problem can be obtained as a limit of solutions to the corresponding problem for1.1when αgoes to zero; see4.
In5, Karch proved well-posedness for a Cauchy problem for the pseudoparabolic 1.1.
2. A Priori Estimates
In this section, we obtain a priori estimates for the problem1.1–1.3.
Lemma 2.1. Letu0 ∈H01Ω. Under assumption1.4, ifuis a solution of the problem1.1–1.3 then one has the following estimate:
∇u2≤D1, A
t
0
∇us2ds≤D2, B
whereD1 >0andD2 >0depend on the initial data and parameters of1.1.
Proof. We multiply1.1byutand integrate overΩ. We get
d
dtEt α∇ut
2
ut20, 2.1
whereEt β/2∇u2−ΩFudx. We integrate2.1on the interval0, t, and from1.4
we get
β 2∇u
2 α
t
0
∇us2ds≤E0. 2.2
3. Continuous Dependence on the Coefficient
α
In this section we prove that the solution of the problem1.1–1.3depends continuously on the coefficientαinH1Ωnorm.
We now assume thatuandvare the solutions of the following problems, respectively:
ut−α1Δut−βΔufu, x∈Ω, t >0,
ux,0 u0x, x∈Ω,
ux, t 0, x∈∂Ω, t >0,
vt−α2Δvt−βΔvfv, x∈Ω, t >0,
vx,0 u0x, x∈Ω,
vx, t 0, x∈∂Ω, t >0.
3.1
Letwu−v,αα1−α2. Thenwis a solution of the problem
wt−α1Δwt−αΔvt−βΔwfu−fv, x∈Ω, t >0, 3.2
wx,0 0, x∈Ω, 3.3
wx, t 0, x∈∂Ω, t >0. 3.4
The following theorem establishes continuous dependence of the solution of1.1–1.3on the coefficientαinH1Ωnorm.
Theorem 3.1. Assume that
fu−fv≤K1 |u|p−1 |v|p−1|u−v|, 3.5
where1< p≤n/n−2ifn >2,p∈1,∞ifn2. Letwbe the solution of the problem3.2–3.4. Thenwsatisfies the estimate
w2 α1∇w2≤Dα1−α22eM1t. 3.6
HereK,D,andM1are positive constants.
Proof. We multiply3.2bywand integrate overΩ. We get
1 2
d dt w
2
α1∇w2
α
Ω∇vt∇wdx β∇w
2
Ω
Using the Cauchy-Schwarz inequality and3.5, we get
d dt
1 2w
2 α1 2 ∇w
2
β∇w2
≤ |α|∇vt∇w K
Ω
1 |u|p−1 |v|p−1|w|2dx.
3.8
Making use of Holder’s inequality, we estimate the second term at the right-hand side of3.8
as follows
K
Ω
1 |u|p−1 |v|p−1|w|2dx≤Kw2 K1
up−1p−1n vp−1p−1n
w2n/n−2w. 3.9
Inequalityw2n/n−2 ≤ K2∇wis valid for allw ∈ H01Ω. Using the Sobolev inequality andA, we obtain the estimate
up−1p−1n vp−1p−1n≤d1
∇up−1 ∇vp−1≤K3. 3.10
Therefore using Poincare’s inequality from3.9and3.10, we get
K
Ω
1 |u|p−1 |v|p−1|w|w dx≤K4∇w2, 3.11
whered1andKii1,2,3,4are positive constants. By using3.11in3.8we get
d dt
1 2w
2 α1 2 ∇w
2
β∇w2≤ |α|∇vt∇w K4∇w2. 3.12
Using arithmetic-geometric mean inequality, we have
d dtE1t≤
|α|2 2 ∇vt
2
M1E1t, 3.13
whereE1t 1/2w2 α1/2∇w2andM1 max{2/α1 K4 1/2,1}.Solving the first-order differential inequality3.13and fromB, we obtain
E1t≤ D2 2 |α|
2
eM1t. 3.14
4. Continuous Dependence on the Coefficient
β
In this section we prove that the solution of the problem1.1–1.3depends continuously on the coefficientβinH1Ωnorm.
We now assume thatuandvare the solutions of the following problems, respectively:
ut−αΔut−β1Δufu, x∈Ω, t >0,
ux,0 u0x, x∈Ω,
ux, t 0, x∈∂Ω, t >0,
vt−αΔvt−β2Δvfv, x∈Ω, t >0,
vx,0 u0x, x∈Ω,
vx, t 0, x∈∂Ω, t >0.
4.1
Letwu−v,ββ1−β2. Thenwis a solution of the problem
wt−αΔwt−β1Δw−βΔvfu−fv, x∈Ω, t >0, 4.2
wx,0 0, x∈Ω, 4.3
wx, t 0, x∈∂Ω, t >0. 4.4
The main result of this section is the following theorem.
Theorem 4.1. Assume that3.5holds. Letw be the solution of the problem4.2–4.4. Thenw
satisfies the estimate
w2 α∇w2≤D1
β1−β2
2
eM2t, 4.5
whereM2is constant.
Proof. We multiply4.2bywand integrate overΩ. We get
1 2
d dt w
2 α∇w2 β
Ω∇v∇w dx β1∇w
2
Ω
fu−fvw dx. 4.6
By using Cauchy-Schwarz inequality and3.5in4.6we get
d dt
1 2w
2 α 2∇w
2
β1∇w2
≤β∇v∇w K
Ω
1 |u|p−1 |v|p−1|w|2dx,
and by using3.11in4.7we obtain
d dt
1 2w
2 α 2∇w
2
β1∇w2≤β∇v∇w K4∇w2. 4.8
Using arithmetic-geometric mean inequality, we have
d dtE2t≤
β2 2 ∇v
2
M2E2t, 4.9
where E2t 1/2w2 α/2∇w2 and M2 max{2/αK4 1/2,1}. Solving the first-order differential inequality4.9and fromA, we obtain
E2t≤ D1 2 β
2
eM2t. 4.10
Hence the proof is completed.
References
1 T. W. Ting, “Certain non-steady flows of second-order fluids,” Archive for Rational Mechanics and Analysis, vol. 14, pp. 1–26, 1963.
2 P. J. Chen and M. E. Gurtin, “On a theory of heat conduction involving two temperatures,”Zeitschrift f ¨ur Angewandte Mathematik und Physik ZAMP, vol. 19, no. 4, pp. 614–627, 1968.
3 R. E. Showalter and T. W. Ting, “Pseudoparabolic partial differential equations,” SIAM Journal on Mathematical Analysis, vol. 1, pp. 1–26, 1970.
4 T. W. Ting, “Parabolic and pseudo-parabolic partial differential equations,”Journal of the Mathematical Society of Japan, vol. 21, pp. 440–453, 1969.
