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On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition
Boundary Value Problems volume 2016, Article number: 137 (2016)
Abstract
We study the following initial-boundary value problem:
where \(\mu>0\), \(\alpha>0\), \(h_{1}\geq0\), \(R>1\) are given constants and f, \(f_{1}\), \(g_{1}\), \(g_{R}\), \(\tilde{u}_{0}\) are given functions. First, we use the Galerkin and compactness method to prove the existence of a unique weak solution \(u(t)\) of Problem (1) on \((0,T)\), for every \(T>0\). Next, we study the asymptotic behavior of the solution \(u(t)\) as \(t\rightarrow+\infty\). Finally, we prove the existence and uniqueness of a weak solution of Problem (1)1,2 associated with a ‘\((N+1)\)-points condition in time’ case,
where \((T_{i},\eta_{i})\), \(i=1,\ldots,N\), are given constants satisfying
1 Introduction
Consider the following nonlinear pseudoparabolic equation:
with the mixed inhomogeneous condition
and with the initial condition
or the ‘\((N+1)\)-points condition in time’ case
where \((T_{i},\eta_{i})\), \(i=1,\ldots,N\), are given constants satisfying
here \(\mu>0\), \(\alpha>0\), \(R>1\), \(h_{1}\geq0\) are given constants and f, \(f_{1}\), \(g_{1}\), \(g_{R}\), \(\tilde{u}_{0}\) are given functions satisfying conditions specified later.
The initial-boundary value problem (1.1)-(1.3) is classical and has a long history of applications and mathematical development. We refer to the monographs of Al’shin [1], and of Carroll and Showalter [2] for references and results on pseudoparabolic or Sobolev type equations. We also refer to [3] for asymptotic behavior and to [4] for nonlinear problems. Problems of this type arise in material science and physics, which have been extensively studied and several results concerning existence, regularity and asymptotic behavior have been established.
Equation (1.1) arises within the frameworks of mathematical models in engineering and physical sciences; see [5–13] and the references therein for interesting results on second grade fluids or a fourth grade fluid or other unsteady flows. It is well known that fluid solid mixtures are generally considered as second grade fluids and are modeled as fluids with variable physical parameters, thus, an analysis is performed for a second grade fluid with space dependent viscosity, elasticity and density.
In [9], some unsteady flow problems of a second grade fluid were considered. The flows are generated by the sudden application of a constant pressure gradient or by the impulsive motion of a boundary. Here, the velocities of the flows are described by the partial differential equations and exact analytic solutions of these differential equations are obtained. Suppose that the second grade fluid is in a circular cylinder and is initially at rest, and the fluid starts suddenly due to the motion of the cylinder parallel to its length. The axis of the cylinder is chosen as the z-axis. Using cylindrical polar coordinates, the governing partial differential equation is
where \(w(r,t)\) is the velocity along the z-axis, ν is the kinematic viscosity, α is the material parameter, and N is the imposed magnetic field. Under the boundary and initial conditions, W is the constant velocity at \(r=a\) and a is the radius of the cylinder.
In [6], two types of time-dependent flows were investigated. An eigen function expansion method was used to find the velocity distribution. The obtained solutions satisfy the boundary and initial conditions and the governing equation. Remarkably some exact analytic solutions are possible for flows involving second grade fluid with variable material properties in terms of trigonometric and Chebyshev functions.
In [5], Mahmood et al. have considered the longitudinal oscillatory motion of a second grade fluid between two infinite coaxial circular cylinders, oscillating along their common axis with given constant angular frequencies \(\Omega_{1}\) and \(\Omega_{2}\). Velocity field and associated tangential stress of the motion were determined by using Laplace and Hankel transforms. In order to find exact analytic solutions for the flow of a second grade fluid between two longitudinally oscillating cylinders, the following problem was studied:
where \(0< R_{1}< R_{2}\), μ, α, \(V_{2}\), \(\Omega_{1}\), \(\Omega_{2}\) are positive constants. The solutions obtained have been presented in the series form in terms of Bessel functions \(J_{0}(x)\), \(Y_{0}(x)\), \(J_{1}(x)\), \(Y_{1}(x)\), \(J_{2}(x)\) and \(Y_{2}(x)\), satisfying the governing equation and all imposed initial and boundary conditions.
The nonlinear parabolic problems of the form (1.1)-(1.3), with/without the term \(( u_{rr}+\frac{\gamma}{r}u_{r} ) \), were also studied in [14, 15] and the references therein. In [14], by using the Galerkin and compactness method in appropriate Sobolev spaces with weight, the authors proved the existence of a unique weak solution of the following initial and boundary value problem for a nonlinear parabolic equation:
Furthermore, asymptotic behavior of the solution as \(t\rightarrow +\infty \) was studied. In [15], the following nonlinear heat equation associated with Dirichlet-Robin conditions was investigated:
The condition (1.3a), which we call ‘\((N+1)\)-points condition in time’, is known as a drifted periodic condition; see [16]. Indeed, if \(u(t)=\sum_{i=1}^{N}\eta_{i}u(t+T_{i})\), in the case of \(0<\vert \eta_{N}\vert \leq1\), then we have
it means
with \(\delta(t)=(\frac{1}{\eta_{N}}-1)u(t)-\frac{1}{\eta_{N}}\sum_{i=1}^{N-1}\eta_{i}u(t+T_{i})\) satisfying the condition
Note that (1.10) holds by the fact that
With \(\eta_{1}=\eta_{2}=\cdots=\eta_{N-1}=0\), \(\eta_{N}=1\), (1.3a) leads to the T-periodic condition
and with \(\eta_{1}=\eta_{2}=\cdots=\eta_{N-1}=0\), \(\eta_{N}=-1\), we have the anti-periodic condition
The present paper is concerned with the second grade fluid in a circular cylinder associated with the initial condition (1.3) or the drifted periodic condition (1.9). The extensive study of such flows is motivated by both their fundamental interest and their practical importance; see [9]. The arrangement of the paper is as follows. In Section 2, we present preliminaries. In Section 3, under appropriate conditions, we prove the existence of a unique weak solution of Problem (1.1)-(1.3). In Section 4, we consider asymptotic behavior of the solution of Problem (1.1)-(1.3), as \(t\rightarrow+\infty\). Finally, in Section 5, we establish the existence and uniqueness of a weak solution of Problem (1.1), (1.2), (1.3a).
Because of the mathematical context, the results obtained here generalize relative to the ones in [14, 15], by using the same techniques and with some appropriate modifications. On the other hand, the fixed point method is also applied.
2 Preliminaries
Put \(\Omega=(1,R)\), \(Q_{T}=\Omega\times(0,T)\), \(T>0\). We omit the definitions of the usual function spaces: \(C^{m} ( \overline {\Omega} ) \), \(L^{p} ( \Omega ) \), \(W^{m,p} ( \Omega ) \). We define \(W^{m,p}=W^{m,p} ( \Omega ) \), \(L^{p}=W^{0,p} ( \Omega ) \), \(H^{m}=W^{m,2} ( \Omega ) \), \(1\leq p\leq \infty\), \(m=0,1,\ldots \) . The norm in \(L^{2}\) is denoted by \(\Vert \cdot \Vert \). We also denote by \((\cdot,\cdot)\) the scalar product in \(L^{2}\). We denote by \(\Vert \cdot \Vert _{X}\) the norm of a Banach space X and by \(X^{\prime}\) the dual space of X. We denote by \(L^{p}(0,T;X)\), \(1\leq p\leq\infty\), the Banach space of the real functions \(u:(0,T)\rightarrow X\) measurable, such that
and
On \(H^{1}\), we shall use the following norm:
We put
V is a closed subspace of \(H^{1}\) and on V two norms \(\Vert v\Vert _{H^{1}}\) and \(\Vert v_{x}\Vert \) are equivalent norms.
Note that \(L^{2}\), \(H^{1}\) are also the Hilbert spaces with the corresponding scalar products
respectively. The norms in \(L^{2}\) and \(H^{1}\) induced by the corresponding scalar products are denoted by \(\Vert \cdot \Vert _{0}\) and \(\Vert \cdot \Vert _{1}\), respectively. V is continuously and densely embedded in \(L^{2}\). Identifying \(L^{2}\) with \((L^{2})^{\prime}\) (the dual of \(L^{2}\)), we have \(V\hookrightarrow L^{2}\hookrightarrow V^{\prime}\); On the other hand, the notation \(\langle\cdot,\cdot \rangle\) is used for the pairing between V and \(V^{\prime}\).
We then have the following lemmas, the proofs of which can be found in [17].
Lemma 2.1
We have the following inequalities:
Lemma 2.2
The imbedding \(H^{1}\hookrightarrow C^{0}( \overline{\Omega})\) is compact.
Lemma 2.3
The imbedding \(V\hookrightarrow C^{0}(\overline{\Omega})\) is compact and
Remark 2.1
On \(L^{2}\), two norms \(v\longmapsto \Vert v\Vert \) and \(v\longmapsto \Vert v\Vert _{0}\) are equivalent. So are two norms \(v\longmapsto \Vert v\Vert _{H^{1}} \) and \(v\longmapsto \Vert v\Vert _{1}\) on \(H^{1}\), and four norms \(v\longmapsto \Vert v\Vert _{H^{1}}\), \(v\longmapsto \Vert v\Vert _{1}\), \(v\longmapsto \Vert v_{x}\Vert \) and \(v\longmapsto \Vert v_{x}\Vert _{0}\) on V.
Consider \(a(\cdot,\cdot)\) is the symmetric bilinear form on \(V\times V\) defined by
with \(h_{1}\geq0\) is given constant.
Then the symmetric bilinear form \(a(\cdot,\cdot)\) is continuous on \(V\times V\) and coercive on V.
We have also the following lemma.
Lemma 2.4
There exists the Hilbert orthonormal base \(\{w_{j}\}\) of \(L^{2}\) consisting of the eigenfunctions \(w_{j}\) corresponding to the eigenvalue \(\bar{\lambda}_{j}\) such that
Furthermore, the sequence \(\{w_{j}/\sqrt{\bar{\lambda}_{j}}\}\) is also the Hilbert orthonormal base of V with respect to the scalar product \(a(\cdot,\cdot)\).
On the other hand, we have \(w_{j}\) satisfying the following boundary value problem:
The proof of Lemma 2.4 can be found in [18], p.87, Theorem 7.7, with \(H=L^{2}\) and \(a(\cdot,\cdot)\) as defined by (2.6).
3 The existence and the uniqueness
Now, we shall consider Problem (1.1)-(1.3) with \(\alpha>0\), \(\mu>0\), \(h_{1}\geq0\) are constants and make the following assumptions:
- (H1):
-
\(\tilde{u}_{0}\in V\);
- (H2):
-
\(g_{1}, g_{R}\in W^{1,1}(0,T)\), \(\tilde{u}_{0x}(1)-h_{1}\tilde{u}_{0}(1)=g_{1}(0)\), \(\tilde{u}_{0}(R)=g_{R}(0)\);
- (H3):
-
\(f_{1}\in L^{1}(0,T;L^{2})\);
- (H4):
-
\(f\in C^{0}(\mathbb{R}; \mathbb{R})\) satisfies the condition that there exists positive constant δ such that
$$(y-z) \bigl( f(y)-f(z) \bigr) \geq-\delta \vert y-z\vert ^{2}\quad \text{for all }y, z\in \mathbb{R}. $$
In the case \(g_{1}\neq0 \) or \(g_{R}\neq0\), it is clearly that Problem (1.1)-(1.3) reduces to a problem with homogeneous boundary conditions by the suitable transformation. Indeed, put
By the transformation \(v(x,t)=u(x,t)-\varphi(x,t)\), Problem (1.1)-(1.3) becomes the following problem:
where
and \(\tilde{u}_{0}\), \(g_{1}\), \(g_{R}\) satisfying the condition \(\tilde {u}_{0x}(1)-h_{1}\tilde{u}_{0}(1)=g_{1}(0)\), \(\tilde{u}_{0}(R)=g_{R}(0)\).
Remark 3.1
The weak formulation of the initial-boundary value problem (3.1) can be given in the following manner: Find \(v\in L^{\infty}(0,T;V)\) with \(tv_{t}\in L^{2}(0,T;V)\), such that v satisfies the following variational equation:
where \(a(\cdot,\cdot)\) is the symmetric bilinear form on \(V\times V\) defined by (2.6).
Then we have the following theorem.
Theorem 3.1
Let \(T>0\) and (H1)-(H4) hold. Then Problem (3.1) has a unique weak solution v such that
Moreover, if (H3) is replaced by \(f_{1}\in L^{2}(Q_{T})\), then the solution v satisfies
Proof
The proof consists of several steps.
Step 1. The Faedo-Galerkin approximation (introduced by Lions [19]).
Consider the basis \(\{w_{j}\}\) for V as in Lemma 2.4. We find the approximate solution of Problem (3.1) in the form
where the coefficients \(c_{mj}\) satisfy the system of nonlinear differential equations
and
The system of equations (3.7) can be rewritten in form
It is clear that for each m there exists a solution \(v_{m}(t)\) in the form (3.6) which satisfies (3.7) almost everywhere on \(0\leq t\leq \tilde{T}_{m}\) for some \(\tilde{T}_{m}\), \(0<\tilde{T}_{m}\leq T\). The following estimates allow one to take \(\tilde{T}_{m}=T\) for all m.
Step 2. A priori estimates.
(a) The first estimate. Multiplying the jth equation of (3.7) by \(c_{mj}(t)\) and summing up with respect to j, afterward, integrating by parts with respect to the time variable from 0 to t, we get after some rearrangements
By \(v_{0m}\rightarrow\tilde{v}_{0}\) strongly in V, we have
where \(C_{0}\) always indicates a bound depending on \(\tilde{v}_{0}\).
By the assumptions (H4), and with \(\varepsilon_{1}>0\), we estimate without difficulty the following terms in (3.10):
Hence, it follows from (3.10)-(3.13) that
where
By Gronwall’s lemma, we obtain from (3.14)
for all \(m\in \mathbb{N}\), for all t, \(0\leq t\leq\tilde{T}_{m}\leq T\), i.e., \(\tilde{T}_{m}=T\), where \(C_{T}\) always indicates a bound depending on T.
(b) The second estimate. Multiplying the jth equation of the system (3.7) by \(2t^{2}c_{mj}^{\prime}(t)\) and summing up with respect to j, we have
Integrating (3.17), we get
We shall estimate the terms of (3.18) as follows:
Note that
hence
This implies
It follows from (3.18)-(3.20) and (3.22) that
for all \(m\in \mathbb{N}\), for all \(t\in{}[0,T]\), \(\forall T>0\), where \(C_{T}\) always indicates a bound depending on T.
By \((tv_{mx})^{\prime}=tv_{mx}^{\prime}+v_{mx}\) and (3.16) we deduce that
Step 3. The limiting process.
By (3.16), (3.23) and (3.24) we deduce that there exists a subsequence of \(\{v_{m}\}\), still denoted by \(\{v_{m}\}\) such that
Using a compactness lemma ([19], Lions, p.57) applied to (3.25), we can extract from the sequence \(\{v_{m}\}\) a subsequence, still denoted by \(\{v_{m}\}\), such that
By the Riesz-Fischer theorem, we can extract from \(\{v_{m}\}\) a subsequence, still denoted by \(\{v_{m}\}\), such that
Because f is continuous, then
On the other hand, by (H4), it follows from (3.21) that
where \(C_{T}\) is a constant independent of m.
Using the dominated convergence theorem, (3.28), and (3.29) yield
Passing to the limit in (3.7) by (3.8), (3.25), (3.30), we have
Step 4. Uniqueness of the solution.
First, we shall need the following lemma.
Lemma 3.2
Let v be the weak solution of the following problem:
Then
Furthermore, if \(\tilde{v}_{0}=0\) then the equality in (3.33) follows.
Lemma 3.2 is a slight improvement of a lemma used in [14] (see also Lions’ book [19]).
Now, we will prove the uniqueness of the solution.
Let \(v_{1}\) and \(v_{2}\) be two weak solutions of (3.1). Then \(v=v_{1}-v_{2}\) is a weak solution of Problem (3.32) with the right-hand side function replaced by \(\tilde{f}(x,t)=-f(v_{1}+\varphi )+f(v_{2}+\varphi )\) and \(\tilde{v}_{0}=0\). Using Lemma 3.2, we get
By (H4), we obtain
It follows from (3.34), (3.35) that
By Gronwall’s lemma \(v=v_{1}-v_{2}=0\).
Assume now that (H3) is replaced by \(f_{1}\in L^{2}(Q_{T})\), then we only show that \(\{v_{m}^{\prime}\}\) is bounded in \(L^{2}(0,T;V)\).
Indeed, multiplying the jth equation of (3.7) by \(c_{mj}^{\prime}(t)\) and summing up with respect to j, afterward, integrating with respect to the time variable from 0 to t, we get after some rearrangements
By the same estimates as above, we obtain
This implies
Then the sequence \(\{v_{m}^{\prime}\}\) is bounded in \(L^{2}(0,T;V)\).
Applying a similar argument used in the proof of Theorem 3.1, the limit v of the sequence \(\{v_{m}\}\) in suitable function spaces, is a unique weak solution of Problem (3.1) satisfying (3.5).
Therefore, Theorem 3.1 is proved. □
4 Asymptotic behavior of the solution as \(t\rightarrow +\infty\)
In this part, let \(T>0\), (H1)-(H4) hold. Then there exists a unique solution \(u=v+\varphi\) of Problem (1.1)-(1.3) such that
We shall study asymptotic behavior of the solution \(u(t)\) as \(t\rightarrow +\infty\).
We make the following supplementary assumptions on the functions \(f_{1} ( x,t ) \), \(g_{1}(t)\), \(g_{R}(t)\):
- (\(\mathrm{H}_{2}^{\prime}\)):
-
\(g_{1}, g_{R}\in W^{1,1}( \mathbb{R}_{+})\), \(\tilde{u}_{0x}(1)-h_{1}\tilde{u}_{0}(1)=g_{1}(0)\), \(\tilde{u}_{0}(R)=g_{R}(0)\), there exist the positive constants \(\bar{C}_{1}\), \(\bar {C}_{R}\), \(\bar{\gamma}_{1}\), \(\bar{\gamma}_{R}\), such that
$$\bigl\vert g_{i}(t)\bigr\vert +\bigl\vert g_{i}^{\prime}(t) \bigr\vert \leq\bar{C}_{i}e^{-\bar {\gamma}_{i}t},\quad \forall t\geq0, i\in \{1, R\}; $$ - (\(\mathrm{H}_{3}^{\prime}\)):
-
\(f_{1}\in L^{\infty}(0,\infty ;L^{2})\), there exist the positive constants \(C_{1}\), \(\gamma_{1}\) and the function \(f_{1\infty}\in L^{2}\), such that
$$\bigl\Vert f_{1}(t)-f_{1\infty }\bigr\Vert _{0} \leq C_{1}e^{-\gamma_{1}t}\quad \forall t\geq0; $$ - (\(\mathrm{H}_{4}^{\prime}\)):
-
\(f\in C^{0}(\mathbb{R};\mathbb{R})\) satisfies the condition that there exists a positive constant δ, with \(0<\delta<\frac{2\mu}{R(R-1)^{2}}\), such that
$$(y-z) \bigl( f(y)-f(z) \bigr) \geq-\delta \vert y-z\vert ^{2}\quad \text{for all }y, z\in \mathbb{R}. $$
First, we consider the following stationary problem:
The weak solution of problem (4.1) is obtained from the following variational problem:
Find \(u_{\infty}\in V\) such that
for all \(w\in V\), where \(a(\cdot,\cdot)\) is the symmetric bilinear form on \(V\times V\) defined by (2.6).
We then have the following theorem.
Theorem 4.1
Let (\(\mathrm{H}_{3}^{\prime}\)), (\(\mathrm{H}_{4}^{\prime}\)) hold. Then there exists a unique solution \(u_{\infty}\) of the variational problem (4.2) such that \(u_{\infty}\in V\).
Proof
Consider the basis \(\{w_{j}\}\) for V as in Lemma 2.4. Put
where \(d_{mj}\) satisfy the following nonlinear equation system:
By Brouwer’s lemma (see Lions [19], Lemma 4.3, p.53), it follows from the hypotheses (\(\mathrm{H}_{3}^{\prime}\)), (\(\mathrm{H}_{4}^{\prime}\)) that system (4.3), (4.4) has a solution \(y_{m}\).
Multiplying the jth equation of system (4.4) by \(d_{mj}\), then summing up with respect to j, we have
By using (H4), we obtain
By using the inequalities (2.5)(iii), (4.6), we obtain from (4.5)
By \(0<\delta<\frac{2\mu}{R(R-1)^{2}}\), choose \(\varepsilon_{1}>0\) such that \(0<\delta+2\varepsilon_{1}<\frac{2\mu}{R(R-1)^{2}}\).
Hence, we deduce from (4.7) that
C̃ is a constant independent of m.
By means of (4.8) and Lemma 2.3, the sequence \(\{y_{m}\}\) has a subsequence still denoted by \(\{y_{m}\}\) such that
On the other hand, by (4.9)2 and the continuity of f, we have
Passing to the limit in equation (4.4), we find without difficulty from (4.9), (4.10) that \(u_{\infty}\) satisfies the equation
Equation (4.11) holds for every \(j=1,2,\ldots \) , i.e., (4.2) holds.
The solution of Problem (4.2) is unique, which can be showed by the same arguments as in the proof of Theorem 3.1.
This completes the proof of Theorem 4.1. □
Now we consider asymptotic behavior of the solution \(u(t)\) as \(t\rightarrow +\infty\).
We then have the following theorem.
Theorem 4.2
Let (H1), (\(\mathrm{H}_{2}^{\prime }\))-(\(\mathrm{H}_{4}^{\prime}\)) hold. Let f satisfy the following condition, in addition:
Then we have
where \(\gamma>0\), \(\bar{C}>0\) are constants independent of t.
Proof
Put \(Z_{m}(t)=v_{m}(t)-y_{m}\). Let us subtract (4.4) from (3.7)1 to obtain
By multiplying (4.14)1 by \(c_{mj}(t)-d_{mj}\) and summing up in j, we obtain
By the assumptions (H1)-(H4), (\(\mathrm{H}_{2}^{\prime}\))-(\(\mathrm{H}_{4}^{\prime}\)), (\(\mathrm{H}_{4}^{\prime\prime}\)) and using the inequality (2.5)(iii), and with \(\varepsilon_{2}>0\), we estimate without difficulty the following terms in (4.15):
(i) Estimate \(\langle f(v_{m}(t)+\varphi(t))-f(y_{m}+\varphi (t)),Z_{m}(t)\rangle\):
(ii) Estimate \(\langle f(y_{m}+\varphi (t))-f(y_{m}),Z_{m}(t)\rangle\).
Note that from the inequalities
and (\(\mathrm{H}_{4}^{\prime\prime}\)), we deduce that
where
Hence
Thus
(iii) Estimate \(\langle f_{2}(t)-f_{1}(t),Z_{m}(t)\rangle\).
We have
Hence
It follows that
Thus
(iv) Estimate \(\langle f_{1}(t)-f_{1\infty},Z_{m}(t)\rangle\).
It follows from (4.15), (4.16), (4.21), (4.25), and (4.26) that
By \(0<\delta<\frac{2\mu}{R(R-1)^{2}}\), choose \(\varepsilon_{2}>0\) such that \(\tilde{\gamma}=\mu- ( \delta+\frac{3\varepsilon _{2}}{2} ) \frac{R}{2}(R-1)^{2}>0\).
Put \(\bar{\gamma}_{0}=\min\{\gamma_{1},\bar{\gamma}_{1},\bar{\gamma}_{R}\}\), we have \(\tilde{\psi}(t)\leq\bar{C}_{0}e^{-2\bar{\gamma}_{0}t}\) for all \(t\geq0\) and
By
where \(\beta_{1}=\frac{1}{2}\min\{\frac{1}{\alpha},\frac {2}{R(R-1)^{2}}\}\).
It follows from (4.28), (4.29) that
Choose \(\gamma>0\) such that \(\gamma<\min\{\bar{\gamma}_{0}, 2\tilde {\gamma}\beta_{1}\}\), then we have from (4.30)
Hence, we obtain from (4.31)
Letting \(m\rightarrow+\infty\) in (4.32) we obtain
or
where
Note that
It follows from (4.34), (4.36) that
This completes the proof of Theorem 4.2. □
5 The existence and uniqueness of a weak solution with respect to \((N+1)\)-points condition in time
In this section, we shall consider Problem (1.1), (1.2), (1.3a) with \(\mu>0\), \(\alpha>0\), \(R>1\), \(h_{1}\geq0\) being given constants and \(T_{i}\), \(\eta_{i}\), \(i=1,\ldots,N\), are given constants satisfying (1.4).
We make the following assumptions:
- (\(\bar{\mathrm{H}}_{2}\)):
-
\(g_{1}, g_{R}\in W^{1,1}(0,T)\), \(g_{1}\), \(g_{R}\) satisfying the \((N+1)\)-points condition in t, i.e.,
$$g_{1}(0)=\sum_{i=1}^{N}\eta _{i}g_{1}(T_{i}), \qquad g_{R}(0)= \sum_{i=1}^{N}\eta _{i}g_{R}(T_{i}); $$ - (\(\bar{\mathrm{H}}_{3}\)):
-
\(f_{1}, f_{1}^{\prime}\in L^{2}(Q_{T})\), \(f_{1}\) satisfying the \((N+1)\)-points condition in time, i.e., \(f_{1}(x,0)=\sum_{i=1}^{N}\eta_{i}f_{1}(x,T_{i})\).
Remark 5.1
An example of the functions \(g_{1}\), \(g_{R}\) satisfying (\(\bar{\mathrm{H}}_{2}\)) are
where \(p>0\), \(\beta_{k}\), \(k\in\{1,R\}\) are constants. It is obvious that (\(\bar{\mathrm{H}}_{2}\)) holds, because
with
and
Similarly, by the transformation \(v(x,t)=u(x,t)-\varphi(x,t)\), with
and by \(\varphi(x,0)=\sum_{i=1}^{N}\eta_{i}\varphi (0,T_{i})\), Problem (1.1), (1.2), (1.3a) reduces to the following problem:
where \(f_{2}(x,t)\) is defined by (3.2)1.
Remark 5.2
The weak formulation of Problem (5.1) can be given in the following manner: Find \(v\in L^{\infty}(0,T;V)\) with \(v_{t}\in L^{2}(0,T;V)\), such that v satisfies the following variational equation:
where \(a(\cdot,\cdot)\) is the symmetric bilinear form on \(V\times V\) defined by (2.6).
Then we have the following theorem.
Theorem 5.1
Let \(T>0\) and (\(\bar{\mathrm{H}}_{2}\)), (\(\bar{\mathrm{H}}_{3}\)), (\(\mathrm{H}_{4}^{\prime}\)) hold. Then Problem (5.1) has a ‘ \((N+1)\)-points condition in time’ weak solution v such that
Furthermore, if \(N=1\), then the solution is unique.
Proof
The proof consists of several steps.
Step 1. The Faedo-Galerkin approximation (introduced by Lions [19]).
Consider the basis \(\{w_{j}\}\) for V as in Lemma 2.4. Let \(W_{m}\) be the linear space generated by \(w_{1},w_{2},\ldots,w_{m}\). We consider the following problem:
Find a function \(v_{m}(t)\) in the form (3.6) satisfying the nonlinear differential equation system (3.7)1 and the \((N+1)\)-points condition in time
We consider the initial value problem given by (3.7), where \(v_{0m}\) is given in \(W_{m}\).
It is clear that for each m, there exists a solution \(v_{m}(t)\) in the form (3.6) which satisfies (3.7) almost everywhere on \(0\leq t\leq\tilde{T}_{m}\) for some \(\tilde{T}_{m}\), \(0<\tilde{T}_{m}\leq T\). The following a priori estimates allow us to take \(\tilde {T}_{m}=T\) for all m.
Step 2. A priori estimates.
Multiplying the jth equation of (3.7)1 by \(c_{mj}(t)\) and summing up with respect to j, we get
By the same estimates as in Section 3, and with \(\varepsilon_{1}>0\), we obtain
Hence, it follows from (5.5)-(5.7) that
By \(0<\delta<\frac{2\mu}{R(R-1)^{2}}\), choose \(\varepsilon_{1}>0\) such that \(\mu- ( \delta+\varepsilon_{1} ) \frac{R}{2}(R-1)^{2}>0\).
Similar to (4.29), we get
where \(\beta_{1}=\frac{1}{2}\min\{\frac{1}{\alpha},\frac {2}{R(R-1)^{2}}\}\).
It follows from (5.8), (5.9) that
where \(\gamma=\beta_{1} [ \mu- ( \delta+\varepsilon _{1} ) \frac{R}{2}(R-1)^{2} ] \), \(f_{\ast}(t)=\frac{1}{\varepsilon_{1}} ( \Vert f_{2}(t)\Vert _{0}^{2}+\Vert f(\varphi (t))\Vert _{0}^{2} ) \).
Integrating (5.10), we have
where \(\rho^{2}=\sup_{0\leq t\leq T}\rho_{1}(t)\), with
Therefore, if we choose \(v_{0m}\) such that \(\Vert v_{0m}\Vert _{0}^{2}+\alpha a(v_{0m},v_{0m})\leq\rho^{2}\), we obtain from (5.11) that
hence
In the space \(W_{m}\) of linear combinations of the functions \(w_{1}, w_{2},\ldots, w_{m}\), we consider the norm \(v_{0m}\longmapsto \Vert v_{0m}\Vert _{\ast}= ( \Vert v_{0m}\Vert _{0}^{2}+\alpha a(v_{0m},v_{0m}) ) ^{1/2}\). Hence
Let \(\bar{B}_{m}(\rho)=\{v_{0m}\in W_{m}:\Vert v_{0m}\Vert _{\ast}\leq\rho\}\) be a closed ball in the space \(W_{m}\). Let us define
We prove that \(\mathcal {F}_{m}\) is a contraction. Let \(v_{0m}, \bar {v}_{0m}\in\bar{B}_{m}(\rho)\) and let \(y_{m}(t)=v_{m}(t)-\bar{v}_{m}(t)\), where \(v_{m}(t)\) and \(\bar{v}_{m}(t)\) are solutions of the system (3.7) on \([0,T]\) satisfying the initial conditions \(v_{m}(0)=v_{0m}\) and \(\bar {v}_{m}(0)=\bar{v}_{0m}\), respectively. Then \(y_{m}(t)\) satisfies the following differential equation system:
\(1\leq j\leq m\), with the initial condition
By using the same arguments as before, we can show that
where \(\bar{\gamma}=\beta_{1} [ \mu-\delta\frac {R}{2}(R-1)^{2} ] >0\), \(\beta_{1}=\frac{1}{2}\min\{\frac{1}{\alpha},\frac {2}{R(R-1)^{2}}\}\).
Integrating the inequality (5.19), we obtain
hence
or
i.e., \(\mathcal {F}_{m}\) is a contraction.
Therefore, there exists a unique function \(v_{0m}\in\bar{B}_{m}(\rho)\) such that the solution of the initial value problem (3.7) is a solution of the system (3.7)1, (5.4). This solution satisfies the inequality (5.13) a.e., in \([0,T]\).
On the other hand, we multiplying the jth equation of (3.7) by \(c_{mj}^{\prime}(t)\) and summing up with respect to j, afterward integrating with respect to the time variable from 0 to T, we get after some rearrangements
From (5.13), we obtain
Note that
and
Hence
This implies
where \(M_{2}(T)=T(R^{2}-1)\sup_{\vert z\vert \leq M_{1}}f^{2}(z)\).
It follows from (5.22), (5.23), (5.24), and (5.28) that
for all \(m\in\mathbb{N}\), for all \(t\in[0,T]\), where \(C_{T}\) always indicates a bound depending on T.
Step 3. The limiting process.
By (5.13) and (5.29) we deduce that there exists a subsequence of \(\{v_{m}\}\), still denoted by \(\{v_{m}\}\) such that
From (5.4), we obtain
Indeed, we prove (5.31) as follows.
By \(\Vert v_{m}(0)\Vert _{\ast}=\Vert v_{0m}\Vert _{\ast}\leq\rho\), and the imbedding \(V\hookrightarrow C^{0}(\overline {\Omega})\) is compact, there exists a subsequence of \(\{v_{0m}\}\), still denoted by \(\{v_{0m}\}\) such that
From the equality \(v_{m}(t)=v_{m}(0)+\int _{0}^{t}v_{m}^{\prime}(s)\,ds\), we deduce from (5.30) and (5.32) that
This implies \(v(0)=\tilde{v}_{0}\) and
From (5.4), we obtain
By (5.30), (5.34) and (5.35), we deduce that
Hence \(v(0)=\sum_{i=1}^{N}\eta_{i}v(T_{i})\), therefore, (5.31) is proved.
Using a compactness lemma ([19], Lions, p.57) applied to (5.30), we can extract from the sequence \(\{v_{m}\}\) a subsequence, still denoted by \(\{v_{m}\}\), such that
By the Riesz-Fischer theorem, we can extract from \(\{v_{m}\}\) a subsequence, still denoted by \(\{v_{m}\}\), such that
Because f is continuous, we have
Using the dominated convergence theorem, (5.27) and (5.38) yield
Denote by \(\{\zeta_{i}, i=1,2,\ldots\}\) the orthonormal base in the real Hilbert space \(L^{2}(0,T)\). The set \(\{\zeta_{i}w_{j}, i, j=1,2,\ldots\}\) forms an orthonormal base in \(L^{2}(0,T;V)\). From (3.7) we have
for all i, j, \(1\leq j\leq m\), \(i\in \mathbb{N}\).
For i, j fixed, we deduce from (5.39) that
Passing to the limit in (5.40) by (5.30), (5.41), we obtain
Equation (5.42) holds for every \(i, j\in \mathbb{N}\), i.e., the equation
is fulfilled.
Step 4. Uniqueness of the solutions.
Assume now that \(N=1\) is satisfied. Let \(v_{1}\) and \(v_{2}\) be two solutions of (5.2). Then \(v=v_{1}-v_{2}\) satisfies the following problem:
Taking \(w=v\) in (5.44)1 and using (5.44)2, we get
Hence
By \(0<\delta<\frac{2\mu}{R(R-1)^{2}}\), implies \(\delta\frac{R}{2}(R-1)^{2}<\mu\), we deduce from (5.46) that \(\int _{0}^{T}a(v(t),v(t))\,dt=0\), i.e., \(v=v_{1}-v_{2}=0\).
This completes the proof of Theorem 5.1. □
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Ngoc, L.T.P., Nhan, T.T., Thuyet, T.M. et al. On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition. Bound Value Probl 2016, 137 (2016). https://doi.org/10.1186/s13661-016-0645-0
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DOI: https://doi.org/10.1186/s13661-016-0645-0