Existence of Time Periodic Solutions for One-Dimensional Newtonian Filtration Equation with Multiple Delays ∗

In this paper, we study one-dimensional Newtonian filtration equation including unbounded sources with multiple delays. The existence of nonnegative non-trivial time periodic solutions will be established by the Leray-Schauder fixed point theorem based on some suitable Lyapunov functionals and some a priori estimates for all possible periodic solutions.


Introduction
Consider the following one-dimensional Newtonian filtration equation with multiple delays subject to the homogeneous Dirichlet boundary value condition where m > 1, a and α are constants, γ is a positive constant, f and g are the known functions satisfying some structure conditions.This kind of equation arises from a variety of areas in applied mathematics, physics and mathematical ecology.For the case m = 1, n = 1 and γ = 0 with f (r) = αr/(1 + r β ), which appears in the blood cell production model [1] On the other hand, for the same case, that is m = 1, n = 1, γ = 0, with different f , the equation (1.1) also is known as the Hematopoiesis model (f (r) = e −kr (k > 0)) as well as Nicholson's blowflies model (f (r) = re −kr (k > 0)), see for example [2,3].While, it is worth noting that all the above models are linearly diffusive, but if nonlinear diffusion is introduced, the model will be more consistent with biologic phenomena in the real world.However, as far as we know, only a few works are concerned with time periodic solutions for degenerate parabolic equation with delay(s).For example, in [4], the authors investigated the existence of time periodic solutions for p-Laplacian with multiple delays.
Nevertheless, in this paper, a more general source will be discussed, which is allowed to be the blood cell production model or other types.
In the present paper, we pay our attention to the existence of nonnegative time periodic solutions for (1.1).It is worth noticing that in the model of [5], the source with delay is a typical but quite special bounded source.However, in this paper, a more general source will be discussed, particularly, the source with delays is allowed to be unbounded, which caused us difficulties in making the maximum norm estimates and some other a priori estimates.On the other hand, the method used in [4] will also not work for the equation we consider, that is the coefficient matrix associated with Lyapunov function depends on solutions of the problem, and therefore the required estimates as did in [4] could not be obtained.So, we must try some other methods.By constructing some suitable Lyapunov functionals, the a priori estimates for all possible periodic solutions, and combining with Leray-Schauder fixed point theorem, we finally establish the existence of time periodic solutions.
The rest of this paper is organized as follows.In Section 2 we introduce some basic assumptions, preliminary lemmas and state the main results of this paper.Section 3 is devoted to investigating the existence of periodic solutions based on the a priori estimates obtained in Section 2 and Leray-Schauder fixed point theorem.EJQTDE, 2010 No. 52, p. 2 Throughout this paper, we make the following assumptions: where T and β i are positive constants, Q = (0, 1) × (0, T ).Since the equation (1.1) is degenerate parabolic and problem (1.1)-(1.2) usually admits solutions only in some generalized sense.Hence we introduce the following definition.
Now we state the main result of this paper.To prove the existence of periodic solutions (1.1)-(1.2),let us first consider the regularized problem The desired solution of the problem (1.1)-(1.2) will be obtained by the limit of some subsequence of solutions u ε of the regularized problem.However, we need first to establish the existence of solutions u ε , for which, we will make use of the Leray-Schauder fixed point theorem and our efforts center on obtaining the uniformly boundness of u ε .To this end, we prove the following lemmas.
satisfying the boundary value condition (2.2), where λ ∈ [0, 1], 0 < ε < 1 is a constant which is arbitrary.Then for any r > 0, we have where C 1 (m, r) > 0 is a constant which depend on m and r.
Proof.Note that, multiplying Eq.(2.4) by u r and integrating over Q, On the other hand .
Then we know that Here and below, we use C > 0 to denote different positive constants depending only on the known quantities.In addition, it is easy to see that , ε 5 and ε 6 are appropriately small, we can get Using Poincaré inequality, we see that which completes the proof of Lemma 2.1.

Lemma 2.3 Assume that (H 1 ), (H 2 ) hold and u is a nonnegative T-periodic solution of the equation (2.4) satisfying the boundary value condition (2.2). Then we have
where C > 0 is a constant.
Proof.The proof is a direct verification.A simple calculation shows For the convenience of further discussion, we denote and have the following result Proof.Define then by the Cauchy inequality and assumption (H 1 ), it follows that EJQTDE, 2010 No. 52, p. 9 where e −α(t−s) u(s)ds. Letting On the other hand Applying Lemma 2.1 and Lemma 2.3 yields Since V is continuous, there exists a t 0 ∈ [0, T ] satisfies Hence, if t 0 ≤ t ≤ t 0 + T , we obtain EJQTDE, 2010 No. 52, p. 12 ≤C.
A simple calculation yields By the definition of V , we see that It follows from the definition of u that The proof is complete.
In addition, we can also easily get the L 2 boundedness of ∂u m ∂t as follows.

Lemma 2.5 Assume that (H 1 ), (H 2 ) hold and u is a nonnegative T -periodic solution of the equation (2.1) satisfying the boundary value condition (2.2). Then there exists a constant
A simple calculation yields Therefore, by Lemma 2.4, we obtain where ε 1 = 2/(Cε).Letting ε 2 be appropriately small, then we can see which completes the proof of Lemma 2.5.We will show the Hölder norm estimate of solutions in the following lemma.

Proof of the Main Result
By means of the above proved lemmas and the Leray-Schauder fixed point theorem, we can obtain the existence of solutions u ε of the regularized problem as follows.
where u − = min{0, u(x, t), (x, t) ∈ Q}.Making use of integration by parts, we have Therefore, u − = 0, a.e. in Q By the definition of u − , we see that u ≥ 0, a.e. in Q.
Consequently, we can rewrite the equation (3.1) as Hence, we know that, if the problem (3.2),(2.2),(2.3)has a T -periodic solution, it must be nonnegative.Similarly, we have the same consequence for the equation with the conditions (2.2) and (2.3), where ζ ≥ 0. On the other hand, with an argument similar to [7], we claim that for any g ∈ C T (Q), the problem EJQTDE, 2010 No. 52, p. 17 x ∈ (0, 1), t ∈ R.
In case that a ≥ 0, we consider the periodic problem of the homotopy equation for regularized problem where for any v(x, t) Then problem (3.4)-(3.5)admits a unique solution u ∈ C α,α/2 T (Q).Define the mapping Since C α,α/2 T (Q) can be compactly embedded into C T (Q), L is compact.By Lemma 2.4, we know that for any fixed point u λ of the mapping L, there is a constant C 0 independent of ε and λ, such that u λ L ∞ ≤ C 0 .
Then in applying Leray-Schauder's fixed point theorem, we know that the problem (2.1)-(2.3)admits a solution u ε .
In case that a < 0, we consider the periodic problem of the homotopy equation for regularized problem ∂u ∂t = ∂ 2 ∂x 2 (εu + u m ) + au + λG(x, t), x ∈ (0, 1), t ∈ R, (3.6) where for any v(x, t) ∈ C T (Q), G(x, t) = f (v(x, t − τ 1 ), • • • , v(x, t − τ n )) + g(x, t) + γ (Q).The following progress is the same as above case, then we get that the problem (2.1)-(2.3)admits a solution u ε .Now, we turn to the proof of the our main result based on the above lemmas and Proposition 3.1 The Proof of Theorem 2.1.Let ε = 1/h (h = 1, 2, • • • ) and we note u h for the solution of the problem (2.2)-(2.4).Clearly, according to Lemma 2.2, Lemma 2.4 and Lemma 2.5, Hence there exists a subsequence {u h } ∞ h=1 , supposed to be {u h } ∞ h=1 itself, and a function u ∈ {u; u ∈ L ∞ ; u m ∈ L ∞ (0, T ; W  Under suitable assumptions, by the similar arguments, the corresponding existence of time periodic solution should be established for evolution equations with variable delays.

2 dxdt
≤ C,where C > 0 is a constant.Proof.In fact, choosing r = m in (2.5), we obtain