Citation: Paul H. Rabinowitz. On a class of reversible elliptic systems[J]. Networks and Heterogeneous Media, 2012, 7(4): 927-939. doi: 10.3934/nhm.2012.7.927
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Several years ago, Bensoussan, Sethi, Vickson and Derzko [1] have been considered the case of a factory producing one type of economic goods and observed that it is necessary to solve the simple partial differential equation
|
$ {−σ22Δzαs+14|∇zαs|2+αzαs=|x|2forx∈RN,zαs=∞as|x|→∞, $
|
(1.1) |
where $ \sigma \in \left(0, \infty \right) $ denotes the diffusion coefficient, $ \alpha \in \left[ 0, \infty \right) $ represents psychological rate of time discount, $ x\in \mathbb{R}^{N} $ is the product vector, $ z: = z_{s}^{\alpha }\left(x\right) $ denotes the value function and $ \left\vert x\right\vert ^{2} $ is the loss function.
Regime switching refers to the situation when the characteristics of the state process are affected by several regimes (e.g., in finance bull and bear market with higher volatility in the bear market).
It is important to point out that, when dealing with regime switching, we can describe a wide variety of phenomena using partial differential equations. In [1], the authors Cadenillas, Lakner and Pinedo [2] adapted the model problem in [1] to study the optimal production management characterized by the two-state regime switching with limited/unlimited information and corresponding to the system
|
$ {−σ212Δus1+(a11+α1)us1−a11us2−ρσ212∑i≠j∂2us1∂xi∂xj−|x|2=−14|∇us1|2,x∈RN,−σ222Δus2+(a22+α2)us2−a22us1−ρσ222∑i≠j∂2us2∂xi∂xj−|x|2=−14|∇us2|2,x∈RN,us1(x)=us2(x)=∞as|x|→∞, $
|
(1.2) |
where $ \sigma _{1}, \sigma _{2}\in \left(0, \infty \right) $ denote the diffusion coefficients, $ \alpha _{1}, \alpha _{2}\in \left[ 0, \infty \right) $ represent the psychological rates of time discount from what place the exponential discounting, $ x\in \mathbb{R}^{N} $ is the product vector, $ u_{r}^{s}: = u_{r}^{s}\left(x\right) $ ($ r = 1, 2 $) denotes the value functions, $ \left\vert x\right\vert ^{2} $ is the loss function, $ \rho \in \left[ -1, 1 \right] $ is the correlation coefficient and $ a_{nm} $ ($ n, m = 1, 2 $) are the elements of the Markov chain's rate matrix, denoted by $ G = [\vartheta _{nm}]_{2\times 2} $ with
| $ \vartheta _{nn} = -a_{nn}\leq 0,\vartheta _{nm} = a_{nm}\geq 0\;{\rm{ and }}\; \vartheta _{nn}^{2}+\vartheta _{nm}^{2}\neq 0\;{\rm{ for }}\;n\neq m, $ |
the diagonal elements $ \vartheta _{nn} $ may be expressed as $ \vartheta _{nn} = -\underset{m\neq n}{\Sigma }\vartheta _{nm} $.
Furthermore, in civil engineering, Dong, Malikopoulos, Djouadi and Kuruganti [3] applied the model described in [2] to the study of the optimal stochastic control problem for home energy systems with solar and energy storage devices; the two regimes switching are the peak and the peak energy demands.
After that, there have been numerous applications of regime switching in many important problems in economics and other fields, see the works of: Capponi and Figueroa-López [4], Elliott and Hamada [5], Gharbi and Kenne [6], Yao, Zhang and Zhou [7] and Wang, Chang and Fang [8] for more details. Other different research studies that explain the importance of regime switching in the real world are [9,10].
In this paper, we focus on the following parabolic partial differential equation and system, corresponding to (1.1)
|
$ {∂z∂t(x,t)−σ22Δz(x,t)+14|∇z(x,t)|2+αz(x,t)=|x|2,(x,t)∈RN×(0,∞),z(x,0)=c+zαs(x),forallx∈RNandfixedc∈(0,∞),z(x,t)=∞as|x|→∞,forallt∈[0,∞), $
|
(1.3) |
and (1.2) respectively
|
$ {∂u1∂t−σ212Δu1+(a11+α1)u1−a11u2−ρσ212∑i≠j∂2u1∂xi∂xj−|x|2=−14|∇u1|2,(x,t)∈RN×(0,∞),∂u2∂t−σ222Δu2+(a22+α2)u2−a22u1−ρσ222∑i≠j∂2u2∂xi∂xj−|x|2=−14|∇u2|2,(x,t)∈RN×(0,∞),(u1(x,0),u2(x,0))=(c1+us1(x),c2+us2(x))forallx∈RNandforfixedc1,c2∈(0,∞),∂u1∂t(x,t)=∂u2∂t(x,t)=∞as|x|→∞forallt∈[0,∞), $
|
(1.4) |
where $ z_{s}^{\alpha } $ is the solution of (1.1) and $ \left(u_{1}^{s}\left(x\right), u_{2}^{s}\left(x\right) \right) $ is the solution of (1.2). The existence and the uniqueness for the case of (1.1) is proved by [10] and the existence for the system case of (1.2) by [11].
From the mathematical point of view the problem (1.3) has been extensively studied when the space $ \mathbb{R}^{N} $ is replaced by a bounded domain and when $ \alpha = 0 $. In particular, some great results can be found in the old papers of Barles, Porretta [12] and Tchamba [13]. More recently, but again for the case of a bounded domain, $ \alpha = 0 $ and in the absence of the gradient term, the problem (1.3) has been also discussed by Alves and Boudjeriou [14]. The interest of these authors [12,13,14] is to give an asymptotic stable solution at infinity for the considered equation, i.e., a solution which tends to the stationary Dirichlet problem associated with (1.3) when the time go to infinity.
Next, we propose to find a similar result as of [12,13,14], for the case of equation (1.3) and system (1.4) that model some real phenomena. More that, our first interest is to provide a closed form solution for (1.3) and (1.4). Our second objective is inspired by the paper of [14,15], and it is to solve the parabolic partial differential equation
|
$ {∂z∂t(x,t)−σ22Δz(x,t)+14|∇z(x,t)|2=|x|2,inBR×[0,T),z(x,T)=0,for|x|=R, $
|
(1.5) |
where $ T < \infty $ and $ B_{R} $ is a ball of radius $ R > 0 $ with origin at the center of $ \mathbb{R}^{N} $.
Let us finish our introduction and start with the main results.
We use the change of variable
|
$ u(x,t)=e−z(x,t)2σ2, $
|
(2.1) |
in
| $ \frac{\partial z}{\partial t}\left( x,t\right) -\frac{\sigma ^{2}}{2}\Delta z\left( x,t\right) +\frac{1}{4}\left\vert \nabla z\left( x,t\right) \right\vert ^{2}+\alpha z\left( x,t\right) = \left\vert x\right\vert ^{2} $ |
to rewrite (1.3) and (1.5) in an equivalent form
|
$ {∂u∂t(x,t)−σ22Δu(x,t)+αu(x,t)lnu(x,t)+12σ2|x|2u(x,t)=0,if(x,t)∈Ω×(0,T)u(x,T)=u1,0,on∂Ω,u(x,0)=e−c+zαs(x)2σ2,forx∈Ω=RN,c∈(0,∞) $
|
(2.2) |
where
|
$ u_{1,0} = \left\{ 1ifΩ=BR,i.e.,|x|=R,T<∞,0ifΩ=RN,i.e.,|x|→∞,T=∞. \right. $
|
Our first result is the following.
Theorem 2.1. Assume $ \Omega = B_{R} $, $ N\geq 3 $, $ T < \infty $ and $ \alpha = 0 $.There exists a unique radially symmetric positive solution
| $ u(x,t)\in C^{2}\left( B_{R}\times \left[ 0,T\right) \right) \cap C\left( \overline{B}_{R}\times \left[ 0,T\right] \right) , $ |
of (2.2) increasing in the time variable and such that
|
$ limt→Tu(x,t)=us(x), $
|
(2.3) |
where $ u_{s}\in C^{2}\left(B_{R}\right) \cap C\left(\overline{B}_{R}\right) $ is the unique positive radially symmetric solution of theDirichlet problem
|
$ {σ22Δus=(12σ2|x|2+1)us,inBR,us=1,on∂BR, $
|
(2.4) |
which will be proved. In addition,
| $ z\left( x,t\right) = -2\sigma ^{2}\left( t-T\right) -2\sigma ^{2}\ln u_{s}\left( \left\vert x\right\vert \right) ,\left( x,t\right) \in \overline{ B}_{R}\times \left[ 0,T\right] , $ |
is the unique radially symmetric solution of the problem (1.5).
Instead of the existence results discussed in the papers of [12,13,14], in our proof of the Theorem 2.1 we give the numerical approximation of solution $ u\left(x, t\right) $.
The next results refer to the entire Euclidean space $ \mathbb{R}^{N} $ and present closed-form solutions.
Theorem 2.2. Assume $ \Omega = \mathbb{R}^{N} $, $ N\geq 1 $, $ T = \infty $, $ \alpha > 0 $ and $ c\in \left(0, \infty \right) $ is fixed. There exists aunique radially symmetric solution
| $ u(x,t)\in C^{2}\left( \mathbb{R}^{N}\times \left[ 0,\infty \right) \right) , $ |
of (2.2), increasing in the time variable and such that
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$ u(x,t)→uαs(x)ast→∞forallx∈RN, $
|
(2.5) |
where $ u_{s}^{\alpha }\in C^{2}\left(\mathbb{R}^{N}\right) $ is the uniqueradially symmetric solution of the stationary Dirichlet problem associatedwith (2.2)
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$ {σ22Δuαs=αuαslnuαs+12σ2|x|2uαs,inRN,uαs(x)→0,as|x|→∞. $
|
(2.6) |
Moreover, the closed-form radially symmetric solution of the problem (1.3) is
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$ z(x,t)=ce−αt+B|x|2+D,(x,t)∈RN×[0,∞),c∈(0,∞), $
|
(2.7) |
where
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$ B=1Nσ2(12Nσ2√α2+4−12Nασ2),D=12α(Nσ2√α2+4−Nασ2). $
|
(2.8) |
The following theorem is our main result regarding the system (1.4).
Theorem 2.3. Suppose that $ N\geq 1 $, $ \alpha _{1}, \alpha _{2}\in \left(0, \infty \right) $ and\ $ a_{11}, a_{22}\in \left[ 0, \infty \right) $ with $ a_{11}^{2}+a_{22}^{2}\neq 0 $. Then, the system (1.4) has a uniqueradially symmetric convex solution
| $ \left( u_{1}(x,t),u_{2}(x,t)\right) \in C^{2}\left( \mathbb{R}^{N}\times \left[ 0,\infty \right) \right) \times C^{2}\left( \mathbb{R}^{N}\times \left[ 0,\infty \right) \right) , $ |
of quadratic form in the $ x $ variable and such that
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$ (u1(x,t),u2(x,t))→(us1(x),us2(x))ast→∞uniformlyforallx∈RN, $
|
(2.9) |
where
| $ \left( u_{1}^{s}(x),u_{2}^{s}(x)\right) \in C^{2}\left( \mathbb{R} ^{N}\right) \times C^{2}\left( \mathbb{R}^{N}\right) $ |
is the radially symmetric convex solution of quadratic form in the $ x $variable of the stationary system (1.2) which exists from the resultof [11].
Our results complete the following four main works: Bensoussan, Sethi, Vickson and Derzko [1], Cadenillas, Lakner and Pinedo [2], Canepa, Covei and Pirvu [15] and Covei [10], which deal with a stochastic control model problem with the corresponding impact for the parabolic case (see [13,16] for details).
To prove our Theorem 2.1, we use a lower and upper solution method and the comparison principle that can be found in [17].
Lemma 2.1. If, there exist $ \overline{u}\left(x\right) $, $ \underline{u}\left(x\right) \in C^{2}\left(B_{R}\right) \cap C\left(\overline{B}_{R}\right) $ two positive functions satisfying
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$ \left\{ −σ22Δ¯u(x)+(12σ2|x|2+1)¯u(x)≥0≥−σ22Δu_(x)+(12σ2|x|2+1)u_(x)inBR,¯u(x)=1=u_(x)on∂BR, \right. $
|
then
| $ \overline{u}\left( x\right) -\underline{u}\left( x\right) \geq 0\mathit{\;for\;all\;}x\in \overline{B}_{R}, $ |
and there exists
| $ u\left( x\right) \in C^{2}\left( B_{R}\right) \cap C\left( \overline{B} _{R}\right) , $ |
a solution of (2.4) such that
| $ \underline{u}\left( x\right) \leq u\left( x\right) \leq \overline{u}\left( x\right) \mathit{\;{\rm{,}}\;}x\in \overline{B}_{R}\mathit{\;{\rm{,}}\;} $ |
where $ \underline{u}\left(x\right) $ and $ \overline{u}\left(x\right) $ arerespectively, called a lower solution and an upper solution of (2.4).
The corresponding result of Lemma 2.1 for the parabolic equations can be found in the work of Pao [18] and Amann [19]. To achieve our goal, complementary to the works [12,13,14,15] it can be used the well known books of Gilbarg and Trudinger [20], Sattinger [17], Pao [18] and a paper of Amann [19]. Further on, we can proceed to prove Theorem 2.1.
By a direct calculation, if there exists and is unique, $ u_{s}\in C^{2}\left(B_{R}\right) \cap C\left(\overline{B}_{R}\right) $, a positive solution of the stationary Dirichlet problem (2.4) then
| $ u\left( x,t\right) = e^{t-T}u_{s}\left( x\right) ,\left( x,t\right) \in \overline{B}_{R}\times \left[ 0,T\right] , $ |
is the solution of the problem (2.2) and
| $ z\left( x,t\right) = -2\sigma ^{2}\left( t-T\right) -2\sigma ^{2}\ln u_{s}\left( x\right) ,\left( x,t\right) \in \overline{B}_{R}\times \left[ 0,T \right] , $ |
is the solution of the problem (1.5) belonging to
| $ C^{2}\left( B_{R}\times \left[ 0,T\right) \right) \cap C\left( \overline{B} _{R}\times \left[ 0,T\right] \right) . $ |
We prove that (2.4) has a unique radially symmetric solution. The existence of solution for (2.4) is obtained by a standard monotone iteration and the lower and the upper solution method, Lemma 2.1. Hence, starting from the initial iteration
| $ u_{s}^{0}\left( x\right) = e^{-\frac{R^{2}-\left\vert x\right\vert ^{2}}{ 2\sigma ^{2}}}, $ |
we construct a sequence $ \{u_{s}^{k}\left(x\right) \}_{k\geq 1} $ successively by
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$ {σ22Δuks(x)=(12σ2|x|2+1)uk−1s(x),inBR,uks(x)=1,on∂BR, $
|
(3.1) |
and this sequence will be pointwise convergent to a solution $ u_{s}\left(x\right) $ of (2.4).
Indeed, since for each $ k $ the right-hand side of (3.1) is known, the existence theory for linear elliptic boundary-value problems implies that $ \{u_{s}^{k}\left(x\right) \}_{k\geq 1} $ is well defined, see [20].
Let us prove that $ \{u_{s}^{k}\left(x\right) \}_{k\geq 1} $ is a pointwise convergent sequence to a solution of (2.4) in $ \overline{B}_{R} $. To do this, first we prove that $ \{u_{s}^{k}\left(x\right) \}_{k\geq 1} $ is monotone nondecreasing of $ k $. We apply the mathematical induction by verifying the first step, $ k = 1 $.
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$ \left\{ σ22Δu1s(x)≤σ22Δu0s(x),inBR,u1s(x)=1=u0s(x),on∂BR. \right. $
|
Now, by the standard comparison principle, Lemma 2.1, we have
| $ u_{s}^{0}\left( x\right) \leq u_{s}^{1}\left( x\right) \;{\rm{ in }}\;\overline{B }_{R}. $ |
Moreover, the induction argument yields the following
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$ u0s(x)=e−R2−|x|22σ2≤...≤uks(x)≤uk+1s(x)≤...in¯BR, $
|
(3.2) |
i.e., $ \{u_{s}^{k}\left(x\right) \}_{k\geq 1} $ is a monotone nondecreasing sequence.
Next, using again Lemma 2.1, we find
|
$ u_s(x):=u0s(x)=e−R2−|x|22σ2≤...≤uks(x)≤uk+1s(x)≤...≤¯us(x):=1in¯BR, $
|
(3.3) |
where we have used
|
$ σ22Δu_s(x)=u_s(x)σ22(|x|2+σ2σ4+N−1σ2)≥u_s(x)(12σ2|x|2+1)σ22Δ¯us(x)=σ22Δ1=0≤¯us(x)(12σ2|x|2+1) $
|
i.e., Lemma 2.1 confirm.Thus, in view of the monotone and bounded property in (3.3) the sequence $\left\{ { u_{s}^{k}\left(x\right)} \right\} $$ _{k\geq 1} $ converges. We may pass to the limit in (3.3) to get the existence of a solution
| $ u_{s}\left( x\right) : = \lim\limits_{k\rightarrow \infty }u_{s}^{k}\left( x\right) \;{\rm{ in }}\;\overline{B}_{R}\;{\rm{,}}\; $ |
associated to (2.4), which satisfies
| $ \underline{u}_{s}\left( x\right) \leq u_{s}\left( x\right) \leq \overline{u} _{s}\left( x\right) \;{\rm{ in }}\;\overline{B}_{R}. $ |
Furthermore, the convergence of $ \left\{ {u_{s}^{k}\left(x\right)} \right\} $ is uniformly to $ u_{s}\left(x\right) $ in $ \overline{B}_{R} $ and $ u_{s}\left(x\right) $ has a radial symmetry, see [15] for arguments of the proof. The regularity of solution $ u_{s}\left(x\right) $ is a consequence of classical results from the theory of elliptic equations, see Gilbarg and Trudinger [20]. The uniqueness of $ u_{s}\left(x\right) $ follows from a standard argument with the use of Lemma 2.1 and we omit the details.
Clearly, $ u\left(x, t\right) $ is increasing in the time variable. The regularity of $ u\left(x, t\right) $ follows from the regularity of $ u_{s}\left(x\right) $. Letting $ t\rightarrow T $ we see that (2.3) holds. The solution of the initial problem (1.5) is saved from (2.1).
Finally, we prove the uniqueness for (2.2). Let
| $ u\left( x,t\right) ,v\left( x,t\right) \in C^{2}\left( B_{R}\times \left[ 0,T\right) \right) \cap C\left( \overline{B}_{R}\times \left[ 0,T\right] \right) , $ |
be two solutions of the problem (2.2), i.e., its hold
|
$ \left\{ ∂u∂t(x,t)−σ22Δu(x,t)+12σ2|x|2u(x,t)=0,if(x,t)∈BR×[0,T),u(x,T)=1,on∂BR, \right. $
|
and
|
$ \left\{ ∂v∂t(x,t)−σ22Δv(x,t)+12σ2|x|2v(x,t)=0,if(x,t)∈BR×[0,T),v(x,T)=1,on∂BR. \right. $
|
Setting
| $ w\left( x,t\right) = u\left( x,t\right) -v\left( x,t\right) \;{\rm{, in }}\; B_{R}\times \left[ 0,T\right] , $ |
and subtracting the two equations corresponding to $ u $ and $ v $ we find
|
$ \left\{ ∂w∂t(x,t)=σ22Δw(x,t)−12σ2|x|2w(x,t),if(x,t)∈BR×[0,T),w(x,T)=0,on∂BR. \right. $
|
Let us prove that $ u\left(x, t\right) -v\left(x, t\right) \leq 0 $ in $ \overline{B}_{R}\times \left[ 0, T\right] $. If the conclusion were false, then the maximum of
| $ w\left( x,t\right) \;{\rm{, in }}\;B_{R}\times \left[ 0,T\right) , $ |
is positive. Assume that the maximum of $ w $ in $ \overline{B}_{R}\times \left[ 0, T\right] $ is achieved at $ \left(x_{0}, t_{0}\right) $. Then, at the point $ \left(x_{0}, t_{0}\right) \in B_{R}\times \left[ 0, T\right) $, where the maximum is attained, we have
| $ \frac{\partial w}{\partial t}\left( x_{0},t_{0}\right) \geq 0\;{\rm{, }}\;\Delta w\left( x_{0},t_{0}\right) \leq 0\;{\rm{, }}\;\nabla w\left( x_{0},t_{0}\right) = 0, $ |
and
| $ 0\leq \frac{\partial w}{\partial t}\left( x_{0},t_{0}\right) = \frac{\sigma ^{2}}{2}\Delta w\left( x_{0},t_{0}\right) -\frac{1}{2\sigma ^{2}}\left\vert x\right\vert ^{2}w\left( x_{0},t_{0}\right) < 0\;{\rm{ }}\; $ |
which is a contradiction. Reversing the role of $ u $ and $ v $ we obtain that $ u\left(x, t\right) -v\left(x, t\right) \geq 0 $ in $ \overline{B}_{R}\times \left[ 0, T\right] $. Hence $ u\left(x, t\right) = v\left(x, t\right) $ in $ \overline{B}_{R}\times \left[ 0, T\right] $. The proof of Theorem 2.1 is completed.
Finally, our main result, Theorem 2.2 will be obtained by a direct computation.
In view of the arguments used in the proof of Theorem 2.1 and the real world phenomena, we use a purely intuitive strategy in order to prove Theorem 2.2.
Indeed, for the verification result in the production planning problem, we need $ z\left(x, t\right) $ to be almost quadratic with respect to the variable $ x $.
More exactly, we observe that there exists and is unique
| $ u\left( x,t\right) = e^{-\frac{h\left( t\right) +B\left\vert x\right\vert ^{2}+D}{2\sigma ^{2}}},\left( x,t\right) \in \mathbb{R}^{N}\times \left[ 0,\infty \right) ,\;{\rm{ with }}\;B\;{\rm{, }}\;D\in \left( 0,\infty \right) \;{\rm{ , }}\; $ |
that solve (2.2), where
|
$ h(0)=c, $
|
(4.1) |
and $ B $, $ D $ are given in (2.8). The condition (4.1) is used to obtain the asymptotic behaviour of solution to the stationary Dirichlet problem associated with (2.2). Then our strategy is reduced to find $ B, D\in \left(0, \infty \right) $ and the function $ h $ which depends of time and $ c\in \left(0, \infty \right) $ such that
| $ - \frac{1}{2}\frac{h^{\prime }\left( t\right) }{\sigma ^{2}}- \frac{\sigma ^{2}}{2}\left[ -\frac{B}{\sigma ^{4}}\left( \sigma ^{2}-B\left\vert x\right\vert ^{2}\right) -\left( N-1\right) \frac{B}{\sigma ^{2}}\right] +\alpha \left( -\frac{h\left( t\right) +B\left\vert x\right\vert ^{2}+D}{2\sigma ^{2}}\right) +\frac{1}{2\sigma ^{2} }\left\vert x\right\vert ^{2} = 0, $ |
or, after rearranging the terms
| $ \left\vert x\right\vert ^{2}\left( 1-\alpha B-B^{2}\right) +N\sigma ^{2}B-\alpha D-h^{\prime }\left( t\right) -\alpha h\left( t\right) = 0, $ |
where (4.1) holds. Now, by a direct calculation we see that the system of equations
|
$ \left\{ 1−αB−B2=0Nσ2B−αD=0−h′(t)−αh(t)=0h(0)=c \right. $
|
has a unique solution that satisfies our expectations, namely,
|
$ u(x,t)=e−ce−αt+B|x|2+D2σ2,(x,t)∈RN×[0,∞), $
|
(4.2) |
where $ B $ and $ D $ are given in (2.8), is a radially symmetric solution of the problem (2.2). The uniqueness of the solution is followed by the arguments in [10] combined with the uniqueness proof in Theorem 2.1. The justification of the asymptotic behavior and regularity of the solution can be proved directly, once we have a closed-form solution. Finally, the closed-form solution in (2.7) is due to (2.1)–(4.2) and the proof of Theorem 2.2 is completed.
One way of solving this system of partial differential equation of parabolic type (1.4) is to show that the system (1.4) is solvable by
|
$ (u1(x,t),u2(x,t))=(h1(t)+β1|x|2+η1,h2(t)+β2|x|2+η2), $
|
(5.1) |
for some unique $ \beta _{1}, \beta _{2}, \eta _{1}, \eta _{2}\in \left(0, \infty \right) $ and $ h_{1}\left(t\right) $, $ h_{2}\left(t\right) $ are suitable chosen such that
|
$ h1(0)=c1andh2(0)=c2. $
|
(5.2) |
The main task for the proof of existence of (5.1) is performed by proving that there exist
| $ \beta _{1},\beta _{2},\eta _{1},\eta _{2},h_{1}\;{\rm{,}}\;h_{2}, $ |
such that
|
$ \left\{ h′1(t)−2β1Nσ212+(a11+α1)[h1(t)+β1|x|2+η1]−a11[h2(t)+β2|x|2+η2]−|x|2=−14(2β1|x|)2,h′2(t)−2β2Nσ222+(a22+α2)[h2(t)+β2|x|2+η2]−a22[h1(t)+β1|x|2+η1]−|x|2=−14(2β2|x|)2, \right. $
|
or equivalently, after grouping the terms
|
$ \left\{ |x|2[−a11β2+(a11+α1)β1+β21−1]−β1Nσ21−a11η2+(a11+α1)η1+h′1(t)+(a11+α1)h1(t)−a11h2(t)=0,|x|2[−a22β1+(a22+α2)β2+β22−1]−β2Nσ22−a22η1+(a22+α2)η2+h′2(t)+(a22+α2)h2(t)−a22h1(t)=0, \right. $
|
where $ h_{1}\left(t\right), $ $ h_{2}\left(t\right) $ must satisfy (5.2). Now, we consider the system of equations
|
$ {−a11β2+(a11+α1)β1+β21−1=0−a22β1+(a22+α2)β2+β22−1=0−β1Nσ21−a11η2+(a11+α1)η1=0−β2Nσ22−a22η1+(a22+α2)η2=0h′1(t)+(a11+α1)h1(t)−a11h2(t)=0h′2(t)+(a22+α2)h2(t)−a22h1(t)=0. $
|
(5.3) |
To solve (5.3), we can rearrange those equations $ 1 $, $ 2 $ in the following way
|
$ {−a11β2+(a11+α1)β1+β21−1=0−a22β1+(a22+α2)β2+β22−1=0. $
|
(5.4) |
We distinguish three cases:
1.in the case $ a_{22} = 0 $ we have an exact solution for (5.4) of the form
|
$ β1=−12α1−12a11+12√α21+a211−4a11(12α2−12√α22+4)+2α1a11+4β2=−12α2+12√α22+4 $
|
2.in the case $ a_{11} = 0 $ we have an exact solution for (5.4) of the form
|
$ β1=−12α1+12√α21+4β2=−12α2−12a22+12√α22+a222−4a22(12α1−12√α21+4)+2α2a22+4 $
|
3.in the case $ a_{11}\neq 0 $ and $ a_{22}\neq 0, $ to prove the existence and uniqueness of solution for (5.4) we will proceed as follows. We retain from the first equation of (5.4)
| $ \beta _{1} = \frac{1}{2}\sqrt{\alpha _{1}^{2}+2\alpha _{1}a_{11}+a_{11}^{2}+4\beta _{2}a_{11}+4}-\frac{1}{2}a_{11}- \frac{1}{2}\alpha _{1}. $ |
and from the second equation
| $ \beta _{2} = \frac{1}{2}\sqrt{\alpha _{2}^{2}+2\alpha _{2}a_{22}+a_{22}^{2}+4\beta _{1}a_{22}+4}-\frac{1}{2}a_{22}- \frac{1}{2}\alpha _{2}. $ |
The existence of $ \beta _{1} $, $ \beta _{2}\in \left(0, \infty \right) $ for (5.4) can be easily proved by observing that the continuous functions $ f_{1}, f_{2}:\left[ 0, \infty \right) \rightarrow \mathbb{R} $ defined by
|
$ f1(β1)=−a11(12√α22+2α2a22+a222+4β1a22+4−12a22−12α2)+(a11+α1)β1+β21−1,f2(β2)=−a22(12√α21+2α1a11+a211+4β2a11+4−12a11−12α1)+(a22+α2)β2+β22−1, $
|
have the following properties
|
$ f1(∞)=∞andf2(∞)=∞, $
|
(5.5) |
respectively
|
$ f1(0)=−a11(12√α22+2α2a22+a222+4−12a22−12α2)−1<0,f2(0)=−a22(12√α21+2α1a11+a211+4−12a11−12α1)−1<0. $
|
(5.6) |
The observations (5.5) and (5.6) imply
|
$ \left\{ f1(β1)=0f2(β2)=0 \right. $
|
has at least one solution $ \left(\beta _{1}, \beta _{2}\right) \in \left(0, \infty \right) \times \left(0, \infty \right) $ and furthermore it is unique (see also, the references [21,22] for the existence and the uniqueness of solutions).
The discussion from cases 1–3 show that the system (5.4) has a unique positive solution. Next, letting
| $ \left( \beta _{1},\beta _{2}\right) \in \left( 0,\infty \right) \times \left( 0,\infty \right) , $ |
be the unique positive solution of (5.4), we observe that the equations $ 3 $, $ 4 $ of (5.3) can be written equivalently as a system of linear equations that is solvable and with a unique solution
|
$ (a11+α1−a11−a22a22+α2)(η1η2)=(β1Nσ21β2Nσ22). $
|
(5.7) |
By defining
|
$ G_{a,\alpha }: = \left( a11+α1−a11−a22a22+α2 \right) , $
|
we observe that
|
$ G_{a,\alpha }^{-1} = \left( α2+a22α1α2+α2a11+α1a22a11α1α2+α2a11+α1a22a22α1α2+α2a11+α1a22α1+a11α1α2+α2a11+α1a22 \right) . $
|
Using the fact that $ G_{a, \alpha }^{-1} $ has all ellements positive and rewriting (5.7) in the following way
|
$ \left( η1η2 \right) = G_{a,\alpha }^{-1} \left( β1Nσ21β2Nσ22 \right) , $
|
we can see that there exist and are unique $ \eta _{1} $, $ \eta _{2}\in \left(0, \infty \right) $ that solve (5.7). Finally, the equations $ 5 $, $ 6 $, $ 7 $ of (5.3) with initial condition (5.2) can be written equivalently as a solvable Cauchy problem for a first order system of differential equations
|
$ {(h′1(t)h′2(t))+Ga,α(h1(t)h2(t))=(00),h1(0)=c1andh2(0)=c2, $
|
(5.8) |
with a unique solution and then (5.1) solve (1.4). The rest of the conclusions are easily verified.
Next, we present an application.
Application 1. Suppose there is one machine producing two products (see [23,24], for details). We consider a continuous time Markov chain generator
|
$ \left( −121212−12 \right) , $
|
and the time-dependent production planning problem with diffusion $ \sigma _{1} = \sigma _{2} = \frac{1}{\sqrt{2}} $ and let $ \alpha _{1} = \alpha _{2} = \frac{1 }{2} $ the discount factor. Under these assumptions, we can write the system (5.4) with our data
|
$ \left\{ β21+β1−12β2−1=0β22−12β1+β2−1=0 \right. $
|
which has a unique positive solution
| $ \beta _{1} = \frac{1}{4}\left( \sqrt{17}-1\right) ,\;{\rm{ }}\;\beta _{2} = \frac{1}{ 4}\left( \sqrt{17}-1\right) . $ |
On the other hand, the system (5.7) becomes
|
$ \left( 1−12−121 \right) \left( η1η2 \right) = \left( β1β2 \right) , $
|
which has a unique positive solution
|
$ η1=43β1+23β2=12(√17−1),η2=23β1+43β2=12(√17−1). $
|
Finally, the system in (5.8) becomes
|
$ \left\{ \begin{array}{l} \left( \begin{array}{c} h_{1}^{\prime }\left( t\right) \\ h_{2}^{\prime }\left( t\right) \end{array} \right) +\left( 1−12−121 \right) \left( h1(t)h2(t) \right) = \left( 00 \right) , \\ h_{1}\left( 0\right) = c_{1}\;{\rm{ and }}\;h_{2}\left( 0\right) = c_{2}, \end{array} \right. $
|
which has the solution
| $ h_{1}\left( t\right) = s_{1}e^{-\frac{1}{2}t}-s_{2}e^{-\frac{3}{2}t}\;{\rm{, }}\; h_{2}\left( t\right) = s_{1}e^{-\frac{1}{2}t}+s_{2}e^{-\frac{3}{2}t}\;{\rm{, with }}\;s_{1},s_{2}\in \mathbb{R}\;{\rm{.}}\; $ |
Next, from
| $ h_{1}\left( 0\right) = c_{1}\;{\rm{ and }}\;h_{2}\left( 0\right) = c_{2}, $ |
we have
|
$ \left\{ s1−s2=c1s1+s2=c2 \right. \Longrightarrow s_{1} = \frac{1}{2}c_{1}+\frac{1}{2}c_{2}\;{\rm{, }}\; s_{2} = \frac{1}{2}c_{2}-\frac{1}{2}c_{1}, $
|
and finally
|
$ \left\{ h1(t)=12(c1+c2)e−12t−12(c2−c1)e−32t,h2(t)=12(c1+c2)e−12t+12(c2−c1)e−32t, \right. \;{\rm{ }}\; $
|
from where we can write the unique solution of the system (1.4) in the form (5.1).
Let us point that in Theorem 2.3 we have proved the existence and the uniqueness of a solution of quadratic form in the $ x $ variable and then the existence of other different types of solutions remain an open problem.
Some closed-form solutions for equations and systems of parabolic type are presented. The form of the solutions is unique and tends to the solutions of the corresponding elliptic type problems that were considered.
The author is grateful to the anonymous referees for their useful suggestions which improved the contents of this article.
The authors declare there is no conflict of interest.
| [1] |
S. Aubry and P. Y. LeDaeron, The discrete Frenkel-Kantorova model and its extensions I-Exact results for the ground states, Physica, 8D, (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6
|
| [2] |
J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21, (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4
|
| [3] | J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier ;(Grenoble), 43, (1993), 1349-1386. |
| [4] | J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. Poincare, Anal. Non Lineaire, 3 (1986), 229-272. |
| [5] | V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré, Anal. Non Lineaire, 6 (1989), 95-138. |
| [6] |
P. Rabinowitz and E. Stredulinsky, "Extensions of Moser-Bangert Theory: Locally Minimal Solutions," Progress in Nonlinear Differential Equations and Their Applications, 81, Birkhauser, Boston, 2011. doi: 10.1007/978-0-8176-8117-3
|
| [7] |
U. Bessi, Many solutions of elliptic problems on $\mathbbR^n$ of irrational slope, Comm. Partial Differential Equations, 30 (2005), 1773-1804. doi: 10.1080/03605300500299992
|
| [8] |
U. Bessi, Slope-changing solutions of elliptic problems on $\mathbbR^n$, Nonlinear Anal., 68 (2008), 3923-3947. doi: 10.1016/j.na.2007.04.031
|
| [9] |
F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in $\mathbbR^{2}$ for a class of non autonomous Allen-Cahn equations, Calc. Var. Partial Differential Equations, 11 (2000), 177-202. doi: 10.1007/s005260000036
|
| [10] |
F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in $\mathbbR^{2}$ for a class of periodic Allen-Cahn equations, Comm. Partial Differential Equations, 27 (2002), 1537-1574. doi: 10.1081/PDE-120005848
|
| [11] |
F. Alessio and P. Montecchiari, Entire solutions in $\mathbbR^{2}$ for a class of Allen-Cahn equations, ESAIM Control Optim. Calc. Var., 11 (2005), 633-672. doi: 10.1051/cocv:2005023
|
| [12] | F. Alessio and P. Montecchiari, Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations, Adv. Nonlinear Stud., 5 (2005), 515-549. |
| [13] |
M. Novaga and E. Valdinoci, Bump solutions for the mesoscopic Allen-Cahn equation in periodic media, Calc. Var. Partial Differential Equations, 40 (2011), 37-49. doi: 10.1007/s00526-010-0332-4
|
| [14] |
R. de la Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg-Landau Allen-Cahn equations in periodic media, Adv. Math., 215 (2007), 379-426. doi: 10.1016/j.aim.2007.03.013
|
| [15] |
R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 26 (2009), 1309-1344. doi: 10.1016/j.anihpc.2008.11.002
|
| [16] |
E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185. doi: 10.1515/crll.2004.068
|
| [17] | D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren, 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983. |
| [18] |
P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 673-688. doi: 10.1016/j.anihpc.2003.10.002
|
| [19] | P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II, Adv. Nonlinear Stud., 4 (2004), 377-396. |
| [20] | S. Bolotin, Existence of homoclinic motions. (Russian), Vestnik Moskov. Univ., Ser. I Mat. Mekh., (1983), 98-103. |
| [21] | A. Anane, O. Chakrone, Z. El Allali and I. Hadi., A unique continuation property for linear elliptic systems and nonresonance problems, Electron. J. Differential Equations, (2001), 20 pp. (electronic). |
| [22] | M. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91. |
| [23] | R. Pederson, On the unique continuation theorem for certain second and fourth order equations, Comm. Pure Appl. Math., 11 (1958), 67-80. |
| [24] |
P. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178
|
| [25] | P. Rabinowitz, A note on a class of reversible Hamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823. |
| [26] |
T. Maxwell, Heteroclinic chains for a reversible Hamiltonian system, Nonlinear Analysis, T.M.A., 28 (1997), 871-887. doi: 10.1016/0362-546X(95)00193-Y
|
| [27] | R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York and London, 1975. |
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Paul H. Rabinowitz. On a class of reversible elliptic systems[J]. Networks and Heterogeneous Media, 2012, 7(4): 927-939. doi: 10.3934/nhm.2012.7.927
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