Time Periodic Solutions for a Viscous Diffusion Equation with Nonlinear Periodic Sources ∗

In this paper, we prove the existence of nontrivial nonnegative classical time periodic solutions to the viscous diffusion equation with strongly nonlinear periodic sources. Moreover, we also discuss the asymptotic behavior of solutions as the viscous coefficientk tends to zero.


Introduction
This paper deals with the following viscous diffusion equation in one spatial dimension under the homogeneous boundary conditions and the time periodic condition u(x, t + ω) = u(x, t), (x, t) ∈ Q, where q > 1, D = ∂/∂x, k > 0 denotes the viscous coefficient, m(x, t) ∈ C 1 (Q) is a positive function satisfies m(x, t + ω) = m(x, t) for any (x, t) ∈ Q, ω is a positive constant.The purpose of this paper is to investigate the solvability of the time periodic problem (1.1)-(1.3)and the asymptotic behavior of solutions as the viscous coefficient k tends to zero.Equations of the form (1.1) can also be called pseudo-parabolic equations [1,2], or Sobolev type equations [3,4].They model many mathematical and physical phenomena, such as the seepage of homogeneous fluids through a fissured rock [5,6], or the heat conduction involving a thermodynamic temperature θ = u − k∆u and a conductive temperature u [7,8], or the populations with the tendency to form crowds [9,10].Furthermore, according to experimental results, some researchers have recently proposed modifications to Cahn's model which incorporate out-of-equilibrium viscoelastic relaxation effects, and thus obtained this type of equations (see [11]).This paper deals with such equations with strong nonlinear time periodic sources, i.e. q > 1.From the early 19th century so far, diffusion equations have been widely investigated, among them periodic problems have been paid much attention.The researches on second order periodic diffusion equations are extensive, and many profound results have been obtained ([12, 13, 14, 15]).When k = 0, i.e. there isn't any viscosity, the equation (1.1) in multi-spatial dimension becomes ∂u ∂t = ∆u + m(x, t)u q , which has been studied in [16,17].The authors proved the existence of nontrivial nonnegative time periodic solutions when q ∈ 1, N N −2 , where N is the spatial dimension.When k > 0, i.e. pseudo-parabolic equations, Matahashi and Tsutsumi established the existence theorems of time periodic solutions for the linear case and the semilinear case [18](1978) and [19](1979), respectively.There are also some other early works that related to the periodic problems of the following well-known BBM equation which also has the viscous term with periodicity conditions with respect to space but not time variable, see for example [20], [21]- [24].As far as we know, there are a few investigations devoted to time periodic problems of this kind of viscous diffusion equations.Furthermore, notice that such equations can be used to describe models which are sensitive to time periodic factors (for example seasons), such as aggregating populations ( [9,10]), etc, and there are some numerical results and the stability of solutions ( [20,25,26]) which indicate that time EJQTDE, 2011 No. 10, p. 2 periodic solutions should exist, so it is reasonable to study the periodic problems of the equation (1.1).This paper is organized as follows.In Section 2, we apply topological degree method to prove the existence of nontrivial nonnegative strong time periodic solutions to the problem (1.1)- (1.3).We further prove that the strong solution is classical.Consequently, in Section 3, we discuss the asymptotic behavior of solutions as the viscous coefficient k tends to zero.

Existence of Periodic Solutions
This section is devoted to the existence of time periodic solutions of the problem (1.1)- (1.3).Due to the time periodicity of the solutions under consideration, we only need to consider the problem on Q ω = (0, 1) ×(0, ω).In fact, the existence results we obtained are finally for the classical solutions, but due to the proof procedure, we first need to define strong solutions of the problem (1.1)-(1.3).
Definition 2.1 Let E = C ω (Q ω ) be the set of all functions which are continuous in [0, 1] × R and ω-periodic with respect to t.A function u is said to be a strong solution Our main result is as follows.
In order to prove this theorem, we employ the topological degree method to get the existence of nontrivial strong time periodic solutions.Finally, by lifting the regularity of the strong solution, we get the classical solution.Actually, the topological degree method enables us to study the problem by considering a simpler equation with parameter where τ ∈ [0, 1] and f ∈ E. Define In the following, we prove that the map F is completely continuous.Furthermore, it is easy to see that if we set f = Φ(u) = m(x, t)|u| q , then the map F (Φ(u), τ ) is also completely continuous.Proof.The existence and uniqueness results can be found in [18].Next, we discuss the regularity of the solutions.Multiplying (2.1) by u and integrating the result with respect to x over (0, 1), by using Young's inequality and Poincaré's inequality, we have here and below, C is a constant independent of u and τ .Then we have d dt Set From (2.4), by the mean value theorem, we see that there exists a point t ∈ (0, ω) such that For any t ∈ ( t, ω], integrating (2.3) from t to t gives Noticing the periodicity of F (t), we arrive Hence, integrating (2.3) over (0, t), we obtain Recalling the definition of F (t) and k > 0, we have Noticing that u(0, t) = 0, there holds Multiplying (2.1) with D 2 u and integrating the result with respect to x over (0, 1), we have Similar to the above discussion, we can obtain ) From the inequality (2.5) and (2.8), we can conclude that We rewrite the equation (2.1) into the following form By using (2.7), (2.10) and recalling k > 0, we get From (2.9), we have For any (x, t 1 ), (x, t 2 ) ∈ Q ω , we consider the case of 0 It follows that Integrating the above equality with respect to y over (x, x + (∆t) β ), from (2.7),(2.12),and by using the mean value theorem, we get where x * = y * + θ * (∆t) β , y * ∈ (x, x + (∆t) β ), θ * ∈ (0, 1).Recalling β ∈ (0, 1), we have (1 − β)/2 ∈ (0, 1/2).Combining the above inequality with (2.13), we have Proof.By Lemma 2.1, the periodicity of u in t, and the Arzelá-Ascoli theorem, we can see that F maps any bounded set of and EJQTDE, 2011 No. 10, p. 6 Denote Similar to the proof of Lemma 2.1, we have Using the method to prove (2.6), we have The proof is complete.Before using the topological degree method, we should remark that if we set f = Φ(u) = m(x, t)|u| q , then the nontrivial strong time periodic solution we obtained are just the nontrivial nonnegative classical solution.
subject to (1.2), (1.3), then u is just the nontrivial nonnegative classical time periodic solution.
Proof.As is well known, (I (2.17) Thus we have Multiplying e t/k on both sides of (2.17), we get For any t ∈ [0, ω], integrating the above equation in [t, t + ω] and using the periodicity of u yield Hence u is the classical solution and we conclude that u ≥ 0. Suppose to the contrary, there exists a pair of points (x 0 , t 0 ) ∈ (0, 1) × (0, ω) such that u(x 0 , t 0 ) < 0.
In the following, we are going to establish the existence of nontrivial strong time periodic solutions by calculating the topological degree.For this purpose, we denote the ball in C(Q ω ) with center zero and radius R by B R (0).Firstly, we calculate deg(I − F (Φ(•), 1), B r (0), 0) for r appropriately small.EJQTDE, 2011 No. 10, p. 8 Proposition 2.2 There exists a constant r > 0 such that Proof.Owing to the complete continuity of the map F (Φ(u), σ), where σ ∈ [0, 1] is a parameter, the homotopy invariance of degree implies Therefore, we need only to prove that there exists a constant r > 0 such that (2.21) holds.
In fact, it suffices to take where m is the upper bound of m(x, t).Suppose u ∈ ∂B r (0), namely u with u and integrating over Q ω , by the time periodicity of u, and noticing that the first eigenvalue of the Laplacian equation with homogeneous Dirichlet boundary value conditions in (0, 1) is π 2 , we have which is a contradiction.The proof is complete.Next, we calculate deg(I − F (Φ(•), 1), B R (0), 0) for appropriately large R.In order to do this, we need the following maximum norm estimate.

Lemma 2.3 If u is a time periodic solution of the equation
subject to the conditions (1.2) and (1.3), then where M 1 is a positive constant independent of u, k and τ .
EJQTDE, 2011 No. 10, p. 9 For proving this lemma, we need the following result, the proof of which is similar to [29].Lemma 2.4 If q > 1, a(x) is appropriately smooth and satisfies 0 < a ≤ a(x) ≤ a, where a and a are positive constants, then the problem where C is constant independent of v. Furthermore, replacing ϕ by ψ in (2.23), we see that Combining the above inequality with (2.24), we obtain .
Then we have v ≡ 0 for any x ∈ R, which is a contradiction.The proof is complete.
Remark 2.1 The result of Lemma 2.4 is also correct for v ∈ H 1 loc (R).
Proof of Lemma 2.3 Suppose that the periodic solution u is not uniformly bounded.Then, there exist unbounded real number collection {ρ n } ∞ n=1 , a sequence {τ n } ∞ n=1 (τ n ∈ [0, 1]) and the periodic solution sequence {u n } ∞ n=1 of the problem (2.22), (1.2) and (1.3), such that Since x n ∈ (0, 1), there exists a subsequence of {x n } ∞ n=1 , denoted by itself for simplicity, and x 0 ∈ (0, 1) such that x n → x 0 as n → ∞.For any fixed n, define where EJQTDE, 2011 No. 10, p. 11 Obviously, v nj L ∞ (Q nj ) = 1 and v nj satisfies here and below, we denote D = ∂ ∂y .Similar to the proof of Lemma 2.1 and Proposition 2.1, we can deduce that v nj ≥ 0. Thus, for simplicity, in what follows, we would throw off the symbol of absolute value of v nj .Therefore, for any φ(y, s) ∈ C 1 (Q nj ) satisfying φ| ∂Ωn = 0, we have Taking φ = v nj , by virtue of the periodicity of v nj , we get Hence, by means of the integral mean value theorem, there exists a point s j ∈ − Noticing that for any s > s j , by taking φ = χ (s j ,s) ∂v nj ∂s , we have where C is a constant independent of j, n and |Ω n |.By the periodicity of v nj , we get Similar to the above argument, we obtain for any s ∈ − . On the other hand, noticing that for any ϕ ∈ C 1 0 (Ω n ), there holds Fixing j 0 > 0, for any j = lj 0 , where l is a positive integer, we get Recalling (2.26), there exists a function as j → ∞.Meanwhile, since mnj is continuous on Q nj 0 , then there exists a function mn , such that mnj → mn as j → ∞.
Hence, taking l → ∞, we have Then, by virtue of the arbitrariness of j 0 , taking j 0 → ∞, we arrive where C is a constant independent of n and R. Therefore, there exists a function v ∈ H 1 loc (R) (pass to a subsequence if necessary) such that as n → ∞.Since mn (y, 0) is continuous on [−R, R], then there exists a function m(y, 0) such that mn (y, 0) → m(y, 0) as n → ∞.Thus, we have and v ≥ 0, ∀y ∈ (−R, R).
Moreover, since v ≡ 0, by the strong maximum principle we have v > 0 for any x ∈ (−R, R).Taking R larger and larger and repeating the argument for the subsequence vk obtained at the previous step, we get a Cantor diagonal subsequence, for simplicity, we still denote it by {v k } ∞ k=1 , which converges in and v > 0, ∀y ∈ R.
Thus, thanks to Lemma 2.4, we see that for q > 1, the above problem has no solution, which is a contradiction.The proof is complete.
In fact, Lemma 2.3 implies that the above inequality holds for R > max{M 1 , r}.
On the other hand, when τ = 0, the equation (2.22) becomes The proof is complete.Finally, we prove our main result of this section.Proof of Theorem 2.1 From Proposition 2.2 and Proposition 2.3, we see that there exist constants R and r satisfying R > r > 0 such that Basing on the discussion in Lemma 2.1 and Proposition 2.1, u is just the nontrivial nonnegative classical time periodic solution.The proof of Theorem 2.1 is complete.

Asymptotic Behavior
In this section, we are interested in the asymptotic behavior of solutions as the viscous coefficient k tends to zero.Here, we denote by C a constant independent of u and k. 3), then u k (x, t) is uniformly convergent in Q ω as k → 0, and the limit function u(x, t) is a nontrivial nonnegative classical periodic solution of the following problem for all (x i , t i ) ∈ Q ω (i = 1, 2), α ∈ (0, 1).Therefore, there exists a function u ∈ H 2,1 (Q ω ) ∩ C α,α/2 (Q ω ) such that as k → 0. Recalling the equation (1.1), we see that for any ϕ ∈ C 2 (Q ω ) satisfying ϕ(x, ω) = ϕ(x, 0) and ϕ(0, t) = ϕ(1, t) = 0 for t ∈ [0, ω], we have

D 2
u k ϕdxdt + Qω m(x, t)u q k ϕdxdt.Taking k → 0, by(3.11),we arrive t)u q ϕdxdt, which implies that u satisfies the equation (3.1) in the sense of distribution.It is obvious that u satisfies the conditions (3.2) and (3.3).Therefore, from the classical theory of the parabolic equation, u(x, t) is a nontrivial nonnegative classical time periodic solution of the problem (3.1)-(3.3).The proof of this theorem is complete.