L∞-energy method for a parabolic system with convection and hysteresis effect

The \begin{document}$L^∞$\end{document} -energy method is developed so as to handle nonlinear parabolic systems with convection and hysteresis effect. The system under consideration originates from a biological model where the hysteresis and convective effects are taken into account in the evolution of species. Some results for the existence of local and global solutions as well as the uniqueness of solution are presented.

The system (1)-(2) could be considered as a biological model where the hysteresis and convective effects are taken into account in the evolution of the species. Equations (1) and (2) correspond to the evolution of 1 + m biological species, for example, prey and m -types of predators, where σ and U = (u 1 , · · · , u m ) are the population densities of the prey and the predators respectively. The hysteresis effect is described by the subdifferential term ∂I U (·) of the indicator function I U (·) of the interval [f * (U ), f * (U )] which depends on the unknown function U . The subdifferential ∂I U (σ) is a set-valued mapping given by where f * , f * : R m → R are given functions such that f * ≤ f * on R m . It is known that the hysteresis effect can be found in many phenomena in nature, for instances, phase transitions, plasticity, ferroelectricity, superconductivity, etc. In particular, there are indications for existence of hysteresis in various biological problems, see e.g., [10], [13]. Although population dynamics is an object of longstanding interest, the mathematical description of hysteresis effect in the processes in population dynamics has been considered only in a few papers. We refer the reader to the paper [7] which seems to be the first paper treating mathematically hysteresis phenomena in a biological problem, namely the authors treated bacterial growth in a petri dish modeled by hysteresis operator of relay type which describes the relation between the rate of the growth of the bacterial population and the pH level of the surrounding acid-buffer mix. The survey paper [14] treats applications of hysteresis in various natural phenomena. One of the chapters of [14] is devoted to the applications in biological problems and the authors of [14] also underscore the necessity of developing of new models for description of hysteresis in biological processes. It is known that some types of hysteresis operators can be represented by ordinary (or partial) differential inclusion containing the subdifferential of the inidicator function of a closed set (whose length/shape could possibly depend on the unknown variables). This fact was pointed out by A. Visintin [24] and was used for analysis of many nonlinear phenomena, for example, the real-time control problems [8], solid-liquid phase transitions [6], [9], shape memory alloy problems [1], filtration problems [12] and this approach was used to some problems arising from population dynamics [2].
In this paper we shall apply L ∞ -energy method which was recently developed (see [22,23,18,19,20]) and found to be an effective tool applicable to various types of parabolic equations and systems including doubly nonlinear parabolic equations, porous medium equations, strongly nonlinear parabolic equations governed by the ∞-Laplacian, etc. The core of the L ∞ -energy method is the derivation of energy estimates in L ∞ even when any energy estimates could not be expected in L p with 1 ≤ p < ∞. The aim of the present paper is to develop L ∞ -energy method so as to solve problems arising from population dynamics with hysteresis effect whose typical example is the system (1)- (4).
The existence of global solutions and the uniqueness of solution of (1)-(4) is already discussed in [16]. The advantage of our treatment based on L ∞ -energy method over [16] lies in the fact that it enables us to obtain easily the a priori bound of the L ∞ -norm of solutions, which leads to the existence of local and global solutions of (1)-(4) under much weaker assumptions on the space dimension N , f * , f * and other given functions. Moreover it also makes it possible to set up a new strategy for the proof of the uniqueness of solution via the uniform L ∞ -estimate of solutions, which ameliorates that in [2] or [9].
The outline of the paper is as follows. In section 2, we fix notations and basic assumptions to state our main results, which claim local and global well-posedness of (1)-(4). In section 3, we prove main results, where we introduce approximate equations with some cut-off technique and establish a priori estimates for solutions of approximate equations with the use of L ∞ -energy method.
2. Preliminary and main results. In this section we formulate our main results. To do this we first prepare some preliminary notes.
For simplicity, we denote by σ and u i the time-derivatives ∂σ ∂t and ∂ui ∂t of σ and u i (i = 1, · · · , m) respectively.
Note that the inclusion (ii) implies that ) for a.e. (x, t) ∈ Q S . We here introduce our basic assumptions: (H3) λ, µ i (i = 1, · · · , m) are locally Lipschitz continuous functions from R into R N , and g and h i (i = 1, · · · , m) are locally Lipschitz continuous functions from R × R m into R. (H4) There exists a positive constant C such that where σ ∈ R, U ∈ R m and |U | = Σ m i=1 |u i | 2 1/2 .

2.2.
Main results. Our main results are stated as follows.
Since the system (1)-(4) has a biological interpretation, the non-negativity of the solution is an important information.
3. Proofs of main results. We here give proofs of main results.
3.1.1. Approximate equations. To prove Theorem 2.2, we rely on "L ∞ -energy method" developed in [18,19,20]. In order to apply L ∞ -energy method, in most cases, we need to introduce approximate equations which admit solutions belonging to L ∞ (Ω). For this purpose, there are several ways of approximation (see [18,19,20]). Here we apply a method to restrict the L ∞ -norm of solution by adding the subdifferential term of some indicator function to the original equation.
To this end, we introduce the restriction parameter M > 0 which will be fixed in the sequel and the indicator function of the closed convex set K M := { u ∈ L 2 (Ω) ; |u(x)| ≤ M a.e. x ∈ Ω } given by Then I M (·) : H = L 2 (Ω) → [0, +∞] becomes a lower semi-continuous convex function and its subdifferential is given by (see [3,4])

Moreover we put
Then ϕ M (·) : H = L 2 (Ω) → [0, +∞] becomes a lower semi-continuous convex function and its subdifferential is given by To see this, it suffices to show that is Lipschitz continuous and monotone increasing in r ∈ R, we get Then Theorem 4.4 and Proposition 2.17 in [4] We further introduce the Yosida approximation ∂I U δ (·) of the hysteresis operator ∂I U (·) for each δ > 0 given below.
Then our approximate equations are given by the following.
We are going to show below that (12)-(15) admits a unique global solution for any σ 0 , u i0 ∈ L ∞ (Ω) ∩ V by applying Corollary IV of [21] with the underlying Hilbert Indeed, we can rewrite the system (12)- (15) in the form of single evolution equation in H as follows: We first note that σ, u i ∈ D(∂ϕ M ) implies that Hence, by virtue of (H2), (H3) and (11), we find that where C M denotes the general constant depending on M . Furthermore (H3) implies (18), (19) and (20), we obtain which assures assumption (A.5) of [21].
The proof of the uniqueness of solution of system (12)-(15) is standard, since λ, µ i , g, h i (i = 1, · · · , m) can be regarded as globally Lipschitz continuous functions by virtue of the boundedness of solutions in L ∞ (Ω).

3.1.2.
A priori estimates. Now let (σ, U ) be the unique solution of the approximate system (12)- (15). We are going to establish some a priori estimates for (σ, U ) which are independent of M and δ.
L ∞ -estimate of U : First, multiply the i-th equation of (13) by |u i | r−2 u i with r ≥ 2 and integrate over Here dividing both sides of (24) by |u i | r−1 L r , we obtain by (25) and (26) Integrating (27) with respect to t over (0, t) and letting r → ∞, we can see by (H3) that there exists a monotone increasing function 1 (·) such that Here Remark 2. Apparently the argument for deriving (27)-(28) from (24)-(26) would not be rigorous, since the meaning of this procedure becomes obscure when the divisor |u i | r−1 L r attains zero. However the following proposition with and k = 1 provides its justification.

Proposition 1. Let y(·) be a nonnegative absolutely continuous function on
where Φ(s, y(s)) is a nonnegative measurable function such that Φ(s, y(s)) ∈ L 1 (0, S), r is a positive integer and k is an integer such that 1 ≤ k < r. Then we have Proof. Let y(s) > 0 for all s ∈ [0, S], then dividing (29) by y(s) r−k > 0, we get a.e. s ∈ (0, S).
Here we prepare the following lemma, which is a generalization of Lemma 2.2 of [20], which corresponds to the case where m 1 (·) ≡ 0 in (42) below.
is absolutely continuous on [0, T 0 ] and it holds that Proof. The assertion follows from the following direct calculation.

Proof of Theorem 2.3: Global solution.
In this subsection, we give a proof for Theorem 2.3. The strategy of the proof is the same as in the previous section. So in order to derive the global solution, it suffices to establish a priori estimates global in t ∈ [0, T ] under (H4). Plugging (6) of (H4) in I 2 given by (26), we get Then dividing both sides of (24) by |u i | r−2 L r (see Remark 2 and apply Proposition 1 with k = 2), we obtain by (25) and (70) Integrating this with respect to t over (0, t) and letting r → ∞, we obtain Now we look at J 3 in (33). Making use of (5) and recalling that u * (t) ≥ 0, we now get Then we obtain We here note that (7) gives (|u * (t)| + |u * (t)|) ≤ C sup 0≤s≤t |U (s)| L ∞ . Thus in parallel with (38), we now have for some constants C 4 and C 5 . Here put then (71) and (74) givê Therefore (75)×(C 4 + 1)+ (76) yieldŝ Thus applying Gronwall's inequality, we can derive the a priori bound for sup 0≤t≤T (|σ(t)| L ∞ + |U (t)| L ∞ ). To complete the proof of Theorem 2.3, it suffices to repeat the same procedures as in the previous section.  (8) with T 0 replaced by S. We denote σ = σ 1 − σ 2 , U = U 1 − U 2 . For simplicity of the notation in the sequel we denote by C various positive constants. We define the auxiliary function L(x, t) by We here claim that In fact, for i = 1 it is clear thatσ x, t)) a.e. in Q S . The estimate from below of (79) is obvious for the case [σ(x, t) − L(x, t)] + = 0. As for the case where [σ(x, t) − L(x, t)] + > 0, we get Thus (79) is verified for i = 1 and the verification for i = 2 can be done analogously. We get two inequalities by putting σ = σ 1 , z =σ 1 and σ = σ 2 , z =σ 2 in (iib) of Definition 2.1. Taking the difference of these two inequalities, we have Hence we obtain 1 2 Denote In order to estimate terms I 1 and I 2 , we need estimates for the derivatives of L. For simplicity of the notation, put Analogously we have Therefore, since H 1 ⊂ L 4 for N ≤ 4, we get a.e. t ∈ (0, S).
As for the terms on the right hand side of (80), we get by (H3) and assumption λ ∈ C 2 (R; R N ) We here put Then in view of (82), we have Then plugging (87) and (88) in (86), we get a.e. t ∈ (0, S).
Thus, by Gronwall's inequality and the fact I(0) = 0, we conclude that I(t) = 0 for t ∈ [0, S] which proves the uniqueness of solution.