GLOBAL FOR LAPLACE REACTION-DIFFUSION EQUATIONS

. We study the initial-boundary value problem for a Laplace reaction-diﬀusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show global existence under suitable assumptions on the reaction function. We also show that the problem generates a dynamical system in a suitably set universal space and that this dynamical system possesses a Lyapunov function.

1. Introduction. We study the initial-boundary value problem for a Laplace reaction-diffusion equation in Ω, (1) in a three-dimensional bounded domain Ω of C 2 class. Here, m(x) is a given function in L ∞ (Ω) such that 0 ≤ m(x) ≤ 1 and m(x) ≡ 0.
(2) The function f (u) is a real valued C 3 function defined for −∞ < u < ∞. It is assumed that with some exponent p ≥ 2 and some constants D i > 0 (i = 1, 2, 3, 4). Note that (3) implies f (0) = 0. On the unknown function u = u(x, t) we impose the homogeneous Dirichlet conditions on the boundary ∂Ω. The initial function u 0 (x) ≥ 0 is a nonnegative function in Ω. Such an elliptic-parabolic equation arises in the study of heat conduction in the composite mediums consisting of several materials that have their own heat conductivity. Let Ω ⊂ R 3 denote such a composite medium and let Ω be divided into the direct sum of subdomains Ω i , 1 ≤ i ≤ n, Ω i denoting a material with a 1474 ANGELO FAVINI AND ATSUSHI YAGI constant heat conductivity a i > 0. Then the equation describing heat conduction in Ω is given by where a(x) is a step function such that a(x) ≡ a i for x ∈ Ω i , 1 ≤ i ≤ n, and where f (u) denotes a nonlinear heat controller. For a test function ϕ(x) ∈ C ∞ 0 (Ω), we have where n i denotes the outer normal vector of ∂Ω i . We here assume on each interface which means that the flux a(x)∇u is continuous on the interface Γ ij and is called the continuity condition on the interface. For the details, see Carslaw-Jaeger [2] and Hahn-Özişik [8]. Under this assumption the heat equation then takes the form in Ω × (0, ∞).
Conversely, if a function u defined in Ω × (0, ∞) satisfies (5) and (7) at the same time, then u is a solution to (5) satisfying the continuity condition (6). That is, the problem of solving (5) under (6) is reduced to that of solving (5) and (7) simultaneously. We want to consider, furthermore, the case where some material may possess extremely larger conductivity than others, say, (for simplicity) a i = ∞ for some i. In such a subdomain, the equation is no longer a heat equation but is a Laplace equation. Then, instead of (7), it is convenient to rewrite the equation into the form where a = min 1≤i≤n a i is a positive number and m(x) is the function a/a(x) for x ∈ Ω. Clearly, m(x) satisfies (2). Under the continuity condition (6), the linear problems were mainly studied until now on the basis of Fourier analysis. Deconinck-Pelloni-Sheils [4] and de Monte [3] construct solutions for one-dimensional linear equations. Mikhailov-Özişik [14] and Salt [15] construct solutions for two and three-dimensional linear equations. Meanwhile, Sheils-Deconinck [16] constructs a mapping from the initial functions to the trace functions of the solutions on the interfaces. We hope that the techniques obtained in this paper together with those of handling (5) will open researches for the nonlinear problems, i.e., (5)- (6).
As for analytical or numerical researches on the general diffusion equations with discontinuous coefficients (not necessarily under the continuity condition on interfaces), we want to quote [1,9,10,11,12,13] and references therein.
First, we construct a unique local solution for (1). Indeed, the local solution can be constructed under rather more general conditions than (3)-(4); for example, C 2 regularity of f (u) is sufficient. We will regard the equation of (1) as a degenerate evolution equation of parabolic type (see (8) below) whose linear problems have been systematically studied by the monograph [7]. Use of the multivalued linear operators introduced in [6] enable us to rewrite the degenerate equation into a multivalued evolution equation (see (14)) but of nondegenerate form. The reduced multivalued evolution equation can then be solved locally by analogous techniques to the usual (single valued) evolution equations. Those are described in the last section of paper.
Second, we show global existence of solutions for suitable initial functions u 0 (x). Under (3)-(4), we establish a priori estimates for local solutions. The reduction of (8) into (14) enables us also to use the theory of infinite-dimensional dynamical systems developed by Temam [17] and others. It is shown that (14) generates a dynamical system whose universal space is suitably determined. It is also shown that every global solution is uniformly bounded and has a nonempty ω-limit set and that there exists a Lyapunov function for this dynamical system, namely, if a global solution is not stationary, then the value of the Lyapunov function decreases strictly as t increases.
Throughout the paper, Ω denotes a C 2 bounded domain in R 3 . For s ≥ 0, H s (Ω) is the complex Sobolev space with exponent s. As usual, H 0 (Ω) = L 2 (Ω). For s > 0, H s 0 (Ω) is the closure of C ∞ 0 (Ω) (space of infinitely differentiable functions in Ω with compact support) in the topology of H 2 (Ω). We shall also use the Sobolev 2. Local solutions. We begin with constructing local solutions to (1) by employing the general theory of semilinear abstract degenerate evolution equations reviewed in Section 5.
2.1. Abstract formulation. Let us formulate (1) as the Cauchy problem for an abstract evolution equation of the form (46), i.e., in the underlying space Y ≡ L 2 (Ω). (9) Here, L is a realization of −a∆ in L 2 (Ω) under the homogeneous Dirichlet conditions on ∂Ω with D(L) ≡ H 2 (Ω) ∩ H 1 0 (Ω). By the estimates of elliptic operators, it is known that Of course, L is a self-adjoint operator of Y . By the Poincaré inequality, there exists a positive constant c such that Hence, L is positive definite in Y and satisfies (47)-(48). According to [18,Theorem 16.12], the domains of its fractional powers L θ are characterized by In particular, we have D(L Meanwhile, the second Banach space X is set by noting that (49) is verified with α = 1 2 (due to (12)). The operator M is then a multiplicative operator by the function m(x) from H 1 0 (Ω) into L 2 (Ω). As verified in [7,Example 3.4], M and L satisfy (50)-(51) with some angle ω < π 2 . Notice that these conditions may fail in L 2 (Ω); so, the settings (9) and (13) are essential.
Finally, f (u) ≡ f (Re u(x)) denotes a nonlinear operator with D(f ) ≡ D(L β ) = H 2β D (Ω) (due to (12)), where β is some fixed exponent such that 3 4 < β < 1. It is known that H 2β (Ω) ⊂ C(Ω) and that u ∈ H 2β D (Ω) if and only if both Re u and Im u belong to H 2β D (Ω) (due to [18,Theorem 1.34]). Then f is a mapping from D(f ) into X. Moreover, since From this it is readily verified that the Lipschitz condition (52) takes place. In this way, all the structural assumptions (47)∼(52) in Section 5 are fulfilled by the three operators L, M and f . The problem (8) is equivalently rewritten in the form in the space X. Here, A ≡ M −1 L is a multivalued linear operator of X given by We fix the third exponentβ in such a way thatβ satisfies 2β − 1 <β < 1.

2.2.
Higher temporal regularities of solutions. In order to obtain the global existence, however, we need higher temporal regularities of solutions. These regularities are obtained by using the techniques established by [5]. In fact, the local solution u actually enjoys: Fix time 0 < t 0 < T . Putting f (t) = f (u(t)) for t 0 ≤ t ≤ T , let us regard u as a solution to the linear equation in L 2 (Ω) on an interval [t 0 , T ]. Then, the derivative of u(t) is naturally expected to satisfies the linear equation in L 2 (Ω) for an unknown function yields that f (t) is an H 1 0 (Ω)-valued Hölder continuous function with exponent θ on the interval [t 0 , T ]. Then, we can apply [7, Theorems 3.10 and 3.11 (α = β = 1)] to (18) with the initial value u (t 0 ) ∈ D(A). So, (18) has a unique solution v in the space: Furthermore, put w(t) = u h (t) − v(t), and observe that w satisfies By [7, Theorem 3.7 (α = β = 1)], w(t) must be represented as This shows that, as h → +0, w(t) → 0 in H 1 0 (Ω) for each t 0 ≤ t < T . Hence, u (t) = v(t) for every t 0 ≤ t < T . By continuity, it is the same at t = T , too. We have thus verified that u lies in the space (19). Since t 0 > 0 is arbitrary, we conclude the desired regularity (17) of u.

Nonnegativity of solutions.
This subsection is devoted to showing nonnegativity of u(t) in Ω under the condition that u 0 ≥ 0 in Ω.
First, u(t) is a real function. Indeed, if u(t) is a solution to (8), then its complex conjugate u(t) is also a solution lying in (16) and having the same initial value. Uniqueness of solution then implies that u(t) = u(t).

ANGELO FAVINI AND ATSUSHI YAGI
In view of this fact, multiply the equation of (8) by H(u(t)) and integrate the product in Ω. Then, which is an open subset of Ω due to (15). Since H (u(x, t)) < 0 for x ∈ Ω − t , ∇u(t) vanishes identically in Ω − t , and hence u(x, t) must be a negative constant in each connected component of Ω − t . Thereby, if Ω − t = ∅, then Ω − t must coincide with Ω because on the boundary of Ω − t , u(x, t) takes negative values. But this contradicts the homogeneous Dirichlet conditions on ∂Ω. Thus, 3. Global solutions. For constructing global solutions, the essential thing is to establish the a priori estimates for local solutions. By the smoothing effect of solutions observed by (16)- (17), there is no loss of generality to assume that u 0 ∈ D(A) = D(M −1 L). Since X is a Hilbert space, the compatibility condition is fulfilled automatically (see (40)). By Remark 2, u is differentiable even at t = 0 and u (0) is determined by the relation (65). We are then led to assume that u is a local solution of (8) lying in Proposition 1. There exists a constant C > 0 such that the estimate holds true for any local solution in (22), the constant C being independent of T u .
Proof. The proof is carried out by several steps.
Step 1. Multiply the equation of (8) by 2u(t) and integrate the product in Ω. Then, it follows by (3) and (20) that Here, we use (11) to obtain that where which will be used later.
Step 2. Similarly, multiply the equation of (8) by 2pu(t) 2p−1 and integrate the product in Ω. Then, after some calculations as above, Solving this differential inequality, we conclude that Step 3. Now, multiply the equation of (8) by 2 ∂u ∂t (t) and integrate the product in Ω. Then, Therefore, by (3), Since we already know (26), it follows that We take a summation of this inequality, (24) and (25). Then, the following differential inequality for ψ 1 Step 4. In view of (22), u t = ∂u ∂t is seen to satisfy Multiply this equation by 2u t and integrate the product in Ω. Then, by (4), After multiplying (27) with a constant 2D 4 + 1, we add the equation to (28). Then, a differential inequality for ψ 2 (t) = ( Hence, As a direct consequence of this estimate and (26), it is observed that L2p + 1]. Hence, since ∇u(t) L2 has already been estimated above, we obtain on account of (10) and (11) that Step 5. As verified in the proof of Theorem 5.1, u(t) satisfies the integral equation Applying the operator Aβ to this equation, we see that Then, by (43), Meanwhile, by (43) and (45), Since it is easily observed by (3), (4) and (15) that yields that f (u(t)) 2 Ultimately, using (15), we obtain that Aβe −(t−s)A f (u(s))ds This jointed with (30) yields the desired estimate (23).
This Proposition 1 readily provides the global existence of solutions.

Moreover, the global solution satisfies the estimates
ψ(·) > 0 being some continuous increasing function.
Proof. First, let us apply Theorem 5.1 to obtain a local solution on an interval [0, T u0 ] in the space (16)- (17). In addition, as verified by (59) and (60), the local solution u satisfies (31) and (32) locally on the interval [0, T u0 ]. Second, we reset a new initial time by the T u0 and a new initial value u(T u0 ). (Note that the condition (21) is satisfied.) Theorem 5.1 again ensures that the original local solution can be extended beyond the time T u0 as an X-valued strict solution, see Remark 2.
Third, we repeat such an extension procedure. Let u be any local solution on an interval [0, T u ]. Applying Theorem 5.1 with initial time T u and with initial value u(T u ), we can extend this local solution to another one on an interval [0, T u + τ ], here τ > 0 is determined by Aβu(T u ) H 1 0 . Since Aβu(t) H 1 0 is uniformly bounded owing to (23) in Proposition 1, the extension length τ is independent of T u ; this means that the original local solution can be extended on the whole interval [0, ∞).
Finally, let us verify the global estimates. As mentioned in Step 1, (31) is the case on the interval [0, T u0 ]. Meanwhile, it is true that Hence, (23) actually means that (31) is the case on the half line [T u0 , ∞).
To verify (32), let 0 < s < ∞ and apply (60) with initial time s and initial value u(s). Then, there is a time length τ > 0 and a constant C u(s) > 0 such that In particular, it is observed that Here, the length τ is determined by the norm Aβu(s) H 1 0 alone. But in view of (31) verified now, that is independent of s. It is the same for C u(s) . Hence, (32) is the case on the half line [ τ 2 , ∞). As mentioned in Step 1, we already know that (32) holds true in the neighborhood of the initial time. 4. Dynamical system. Knowing that (8) has a unique global solution for every initial value u 0 ∈ D(Aβ) satisfying u 0 (x) ≥ 0, where A = M −1 L, let us construct a dynamical system in the underlying space D(Aβ).
Set a phase space by K being equipped with the norm Aβ · X . For u 0 ∈ K, let u(t; u 0 ) be the global solution of (8), and set By Theorem 3.1, S(t) maps K into itself. By uniqueness of solution, S(t) satisfies the semigroup property S(t + s) = S(t)S(s) for 0 ≤ s, t < ∞. Meanwhile, S(t) is locally Lipschitz continuous on K. In fact, put Then, by (31), all the trajectories S(t)u 0 starting from K R remain in a bounded subset of K ψ(R) . In addition, thanks to Theorem 5.3, there is τ R > 0 such that

GLOBAL EXISTENCE FOR LAPLACE REACTION-DIFFUSION EQUATIONS 1483
For 0 < t < ∞, let t = nτ R + t 0 with 0 ≤ t 0 < τ R and some integer n. Then, we have Meanwhile, as X is a Hilbert space, Remark 1 shows that S(t)u 0 is continuous at t = 0 with respect to the graph norm of Aβ. We have thus verified that S(t) is a continuous nonlinear semigroup on K and (S(t), K, D(Aβ)) defines a dynamical system.
The dynamical system is shown to possesses a Lyapunov function. Indeed, for u 0 ∈ K and u(t; u 0 ) = S(t)u 0 , multiply the equation of (8) by ∂u ∂t and integrate the product in Ω. Then, where F (u) = u 0 f (v)dv is a primitive function of f (u). This then shows that the function plays the role of a Lyapunov function to (S(t), K, D(Aβ)).
The following properties of Ψ(·) are verified.
By the standard arguments, the following theorem is proved.
Theorem 4.1. For any u 0 ∈ K, the ω-limit set ω(u 0 ) of the trajectory S(t)u 0 in D(Aβ) is a nonempty set and consists of stationary solutions of (8).
see [6]. When A is a multivalued linear operator, A0 is always a linear subspace of X, and for u ∈ D(A) it is true that Au = f + A0 with any f ∈ Au.
where 0 < ω < π, and that (λ − A) −1 satisfies with some constant D > 0. If 0 ∈ ρ(A), that is A −1 ∈ L(X), the graph Then, under 0 ∈ ρ(A), D(A) becomes a Banach space with the graph norm When X is a reflexive Banach space, it is known that holds automatically for any u ∈ D(A) and any f ∈ X.
When A is sectorial, its fractional powers for negative exponents are defined by the integrals But, since f ∈ A y u is arbitrary, we observe (42) to be true. Let now A be a sectorial operator with angle ω < π 2 . Then the analytic semigroup e −tA generated by −A is given by the integral in L(X), the integral contour Γ being as above, with the norm estimate If f ∈ A0, then e −tA f = 0 for all t > 0; therefore, as t → 0, e −tA f does not converge to f in general. As a matter of fact, it is only verified like (37) that see the second Remark to [7,Theorem 3.5]. It is seen that A x e −tA is single valued for every t > 0, although A x is multivalued. Moreover, A x e −tA satisfies the norm estimate in Y . Here, L is a sectorial operator of Y . That is, L is a densely defined, closed linear operator whose spectrum is contained in an open sectorial domain with 0 < ω < π and whose resolvent satisfies the estimate with some constant D > 0. Meanwhile, M is a bounded linear operator from X into Y , where X is another Banach space with norm · X such that with some 0 ≤ α < 1. It is assumed that M -spectrum of L is contained in an open sectorial domain σ M (L) ⊂ Σ = {λ ∈ C; | arg λ| < ω} (50) with some angle 0 < ω < π 2 , and that the M -resolvent (λM − L) −1 of L satisfies with some constant D > 0. Finally, f is a nonlinear operator from D(f ) (⊃ D(L)) into X. We assume that there is an exponent β such that α ≤ β < 1 for which it holds that D(L β ) ⊂ D(f ) together with the Lipschitz condition where ϕ(·) is some continuous increasing function. It clearly follows that The initial value u 0 is taken in D(L β ). Under these structural assumptions (47)-(52), one can show local existence of strict solution for (46).
Let us fix an exponent 0 <β < 1 so that T u0 > 0 being determined by the norm Aβu 0 X alone.
Moreover, the local solution u satisfies the estimates C u0 > 0 being determined by the norm Aβu 0 X alone.
Proof. For 0 < T < ∞, we set a Banach space X (T ) by equipped with the norm u X = max 0≤t≤T L β u(t) Y . In addition, we set a closed ball B(T ) of X (T ) by where the radius R > 0 will be specified below.
For u ∈ B(T ), we define a mapping Let us verify that, if R is suitably chosen and if T is sufficiently small, then Φ is a contraction of X (T ) which maps B(T ) into itself.
Step 1. The function [Φu](t) is seen to be a Hölder continuous function with values in D(L β ). To show this, we need to introduce an auxiliary exponentβ such that 0 <β <β but β < (1 −β )α +β (see (57)). Then, it follows by Proposition 3 that Let g 0 be any element such that g 0 ∈ Aβu 0 . Then, since u 0 = A −β g 0 , we have Therefore, by [7,Theorem 3.5], we obtain that Then, in view of (45) and (53), the first term in the right hand side is estimated by Similarly, the second term is estimated by Hence, we have observed that In particular, Φ is a mapping from B(T ) into X (T ).
Step 2. Let us verify that Φ can map B(T ) into itself. Using (53) and arguing in a similar way as above, we easily verify that with some positive constants C and C . Then, choose now R in such a way that R = C g 0 X + 1.
Furthermore, diminish T > 0 in such a way that Then, Φ maps B(T ) into itself.
Step 3. In the meantime, Φ can be a contraction of X (T ). In fact, for u, v ∈ B(T ), Therefore, after some computations, This shows that Φ is a contraction provided we further diminish T > 0.
Step 4. By the fixed point theorem for contraction mappings, we conclude that Φ has a unique fixed point u = [Φu] in B(T ). (61) jointed with (52) then implies that f (u) ∈ C σ ([0, T ]; X). Thanks to [7, Theorem 3.7] on the linear multivalued equations, we obtain that u has the regularity u ∈ C 1 ((0, T ]; X) together with u ∈ C((0, T ]; D(A)) and satisfies the multivalued equation of (54). Moreover, according to the second Remark to [7,Theorem 3.7], u (t) is represented as It is then verified that u is a strict solution to (54) belonging to (58). Let us verify the estimates (59) and (60). Since we see that Aβu(t) X ≤ C( g 0 X + 1) by the definition of the graph norm (36). Thereby, we obtain (59). Meanwhile, it follows from (62) that Then, due to (45) with x = 1 −β and 1, As mentioned above, u = Φ(u) and (61) yield that u(t) is Hölder continuous with values in D(L β ) at the exponent σ; therefore, (52) implies that f (u(t)) is Hölder continuous with values in X at the same exponent. Hence, we obtain that which is the first estimate of (60).
Noting that Au(t) −u (t) + f (u(t)), the second one of (60) is also observed. We remember that all the constants appearing in the arguments were determined by the norm g 0 X alone. But, since g 0 was any element of Aβu 0 , it is possible to assert that T and C u0 are determined by Aβu 0 X alone.
Step 5. It remains to see uniqueness of the solution to (54) in the space (58). So, let v be any other solution lying in (58). Then, thanks to [7, Theorem 3.7] again, v(t) must be equal to [Φv](t) for any 0 ≤ t ≤ T u0 . Thereby, For 0 < S ≤ T u0 , we see that This means that, if S > 0 is sufficiently small, then u(t) = v(t) for all 0 ≤ t ≤ S. Consequently, We can repeat this argument to conclude that u(t) = v(t) for all 0 ≤ t ≤ T u0 .
Remark 1. We remark that the solution u may not be continuous at t = 0 with respect to the graph norm of D(Aβ). Indeed, from it is observed that u(t) → u 0 in D(Aβ) if e −tA g 0 → g 0 in X as t → 0. But, in view of (44), this is the case only when g 0 ∈ D(A), i.e., It is however observed that, when X is a reflexive Banach space, this condition is automatically fulfilled for any u 0 ∈ D(Aβ). Indeed, if g 0 ∈ Aβu 0 , then g 0 = f + f with f ∈ A0 and f ∈ D(A) due to (39); since A0 = Aβ0 in general, on one hand, we have g 0 − f ∈ Aβu 0 ; on the other hand, f ∈ D(A). Hence, (63) is the case.
then u is differentiable at t = 0, too, and satisfies the equation of (54) at the initial time. In view of (44) this fact is verified by the second Remark to [7,Theorem 3.7]. As mentioned by (40), when X is a reflexive Banach space, this condition is automatically fulfilled for any u 0 ∈ D(A).
It is immediate to verify that the solution of (54) constructed above gives a unique solution to (46) lying in the function space: In fact, if u is a solution of (54), then it follows from du dt (t)−f (u(t)) ∈ M −1 Lu(t) that M du dt (t) − f (u(t)) = Lu(t). In addition, u naturally belongs to (66). Conversely, if u is a solution to (46) lying in (66), then u actually belongs to (58) and satisfies the multivalued equation of (54).
Next, we notice higher regularities of the local solution. These properties often play an important role (see [5]).
Finally, let us show continuous dependence of local solutions on the initial values. Set a subset of initial values K r = {u 0 ∈ D(Aβ); Aβu 0 X ≤ r} for r > 0. Theorems 5.1 and 5.2 provide existence of local solutions lying in (67) and (68) for every u 0 ∈ K(r) on a unified interval [0, T r ].
Theorem 5.3. There exists a constant C r > 0 such that, for any pair of u 0 , v 0 ∈ K r and their local solutions u, v, respectively, it holds that provided that 0 < T (≤ T R ) is suitably diminished.