Semilinear Parabolic Equations

Let Ω ⊂ IRⁿbe the interior of a compact n-dimensional manifold with C² boundary ∂Ω.

We consider the semilinear system of m coupled equations (1 ≤ i ≤ m):

∂ui

∂t = (Aiui)(t, x) + fi(t, x, u, ∇u), (x ∈ Ω, t > t0) (Biui)(t, x) = 0, (x ∈ ∂Ω, t > t0)

ui(t0, x) = v0,i(x) (x ∈ Ω )









(6.6)

Here the unknown function is u = (u1, . . . , um) : Ω → IR^m, and ∇u := (∇u1, . . . , ∇um) ∈ (IRⁿ)^m lists the spatial gradients ∇ui of the ui, i.e., ∇ui :=

∂ui

∂x1, . . . ,_∂x^∂u_nⁱ

. Each Ai(x) is a second order, elliptic differential operator of the form

Ai(x) = with uniformly continuous and bounded coefficients. Each n×n matrix Cⁱ(x) := [C_ljⁱ(x)]

is assumed positive definite:

0 < inf(hCⁱ(x)y, yi), (x ∈ Ω, y ∈ IRⁿ, |y| = 1) where h·, ·i denotes the Euclidean inner product on IRⁿ.

The function differentiable vector field transverse to ∂Ω and pointing outward from Ω. Note that Neumann is a special case of Robin.

We rewrite (6.6) as an initial-boundary value problem for an unknown vector-valued function u := (u1, . . . , um) : [t0, τ ) × Ω → IR^m, function space on Ω, whose trajectories are solutions to (6.8).

Of special interest are autonomous systems, for which f = f (x, u, ∇u); and the reaction-diffusion systems, characterized by f = f (t, x, u).

Assume n < p < ∞. To Equation (6.8) we associate an abstract differential equation (6.1) in L^p(Ω, IR^m). The pair of operators (Ai, Bi) has a sectorial realization Ai in L^p(Ω) with domain D(Ai) ,→ L^p(Ω) (Lunardi [121], 3.1.3). The operator A := A₁× · · · × Am

is sectorial on X := L^p(Ω, IR^m) = [L^p(Ω)]^m.

For α ∈ [0, 1) set X^α := X^α(A). We choose α so that f defines a continuous substitution operator

F : IR₊× X^α → X, F (t, u)(x) := f (t, x, u(x), ∇u(t, x)) It suffices to take α

1 > α > 1 2

 1 + n

p



, (6.9)

for then X^α ,→ C¹(Ω, IR^m) by the Sobolev embedding theorems.

The data (A, F ) thus determine an abstract differential equation u⁰ = Au + F (t, u) in X, whose trajectories u(t) correspond to solutions u(t, x) := u(t)(x) of (6.6). The assumptions on f make F (t, u) locally Lipschitz in u ∈ X^α.

By Theorem 6.1 and the Sobolev embedding theorem we have:

Proposition 6.8 Equation (6.8) defines a solution process Θ on X := L^p(Ω, IR^m) which induces a solution process in X^β for every β ∈ [0, 1) with β ≥ α.

We quote a useful condition for globality of a solution:

Proposition 6.9 Assume there are constants C > 0 and 0 <  ≤ 1 such that

kf (t, x, v, ξ)k ≤ C(1 + kvk + kξk^2−) for all (t, x, v, ξ) ∈ IR+× Ω × S × IRⁿ (6.10) If u : [t0, τ ) → L^p(Ω, IR^m) is a trajectory such that

lim sup

t→τ −

ku(t)kL^p(Ω,IR^m) < ∞ (6.11) then τ = ∞.

Proof Follows from Amann [9], Theorem 5.3(i), taking the constants of that result to be m = k = p0 = γ0 = 1, κ = s0 = 0, γ1 = 2 − .

Solutions u : [t0, τ ) × Ω → IR^m to (6.8) enjoy considerable smoothness. For ex-ample, if the data ∂Ωi, fi, Ai, Bi are smooth of class C^2+2, 0 < 2 < 1, then u ∈ C^1+,2+2 [t1, t2] × Ω, IR^m

for all t0 < t1 < t2 < τ (Lunardi [121], 7.3.3(iii)).

While useful for many purposes, solution processes in the spaces X^α suffer from the drawback that X^α and its norm are defined implicitly, leaving unclear the domains of solutions and the meaning of convergence, stability, density and similar topological terms. In addition, the topology of X^α might be unsuitable for a given application. To overcome these difficulties we could appeal to results of Colombo and Vespri [29], Lunardi [121] and Mora [143], establishing induced processes in Banach spaces of continuous, smooth or L^p functions; or we can apply Theorem 6.2. We now define these spaces.

For r ∈N let C^r(Ω) denotes the usual Banach space of C^r functions on Ω. Set C₀^r(Ω) := {v ∈ C^r(Ω) : v|∂Ω = 0}

With γ, ξ as in a Robin boundary operator and r ≥ 1, define C_γ,ξ^r (Ω) :=



v ∈ C^r(Ω) : γ(x)v(x) + ∂v

∂ξ(x) = 0, (x ∈ ∂Ω)



It is not hard to show that:

• C⁰(Ω), C¹(Ω) and C_γ,ξ¹ (Ω) are strongly ordered, with u  0 if and only if u(x) > 0 for all x ∈ Ω

• C₀¹(Ω) is strongly ordered, with u  0 if and only if u(x) > 0 for all x ∈ Ω and

∂u/∂ν > 0 where ν : ∂Ω → IRⁿ is the unit vector field inwardly normal to ∂Ω.

• C₀⁰(Ω) is not strongly ordered. Both C₀⁰(Ω) and C⁰(Ω) are Banach lattices.

In terms of the boundary operators Bi, for k = 0, 1 we define Banach spaces

C_B^k_i(Ω) :=







C₀^k(Ω) if Bi is Dirichlet

C_γ,ξ^k (Ω) if Bi is Robin and k = 1 C⁰(Ω) if Bi is Robin and k = 0

Note that C_B¹_i(Ω) is strongly ordered, while C_B⁰_i(Ω) is strongly ordered if and only if Bi

is Robin; C_B⁰_i(Ω) is a Banach lattice. The ordered Banach space C_B^k(Ω, IR^m) := ΠiC_B^k_i(Ω),

with the product order cone, is strongly ordered if k = 1, or k = 0 and no Bi is Dirichlet.

The order cone L^p(Ω, IR^m)+ is the subset of L^p(Ω, IR^m) comprising equivalence classes represented by functions Ω → IR^m₊. Note that L^p(Ω, IR^m) is normally ordered but not strongly ordered.

It is known that the pair of operator (Ai, Bi) has a sectorial realization Ai on C^k(Bi) and therefore the product operator A is sectorial on C_B^k(Ω, IR^m). See Corollary 3.1.24, Theorems 3.1.25, 3.1.26 in [121].

Lemma 6.10 For X = L^p(Ω, IR^m) or C_B^k(Ω, IR^m), the analytic semigroup e^tA is a pos-itive operator for t ≥ 0 with respect to the cone of componentwise nonnegative functions in X.

Proof As noted in Remark 6.7, it suffices to show that (λI − A)⁻¹ is positive for large λ > 0, or equivalently, that for each i and fi ≥ 0, the solution gi ∈ D(Ai) of fi = λgi− Aigi satisfies gi ≥ 0. The existence of gi is not the issue but rather it’s positivity.

Thus it boils down to λgi − Aigi ≥ 0 =⇒ gi ≥ 0. But these follow from standard maximum principle arguments. See Lemma 3.1.4 [151].

With X = L^p(Ω, IR^m) and A and α as above, we have a chain of continuous inclusions of ordered Banach spaces

D(A) ,→ X^α ,→ C_B¹(Ω, IR^m) ,→ C_B⁰(Ω, IR^m) ,→ L^p(Ω, IR^m), with a solution process in L^p(Ω, IR^m) and an induced solution process in X^α.

Proposition 6.11 Let Θ be the solution process in L^p(Ω, IR^m) for Equation (6.1) with n < p < ∞.

(a) For all t > t0, Θt,t0 maps Dt,t0 continuously into C_B¹(Ω, IR^m).

(b) Θ induces a solution process Θ¹ in C_B¹(Ω, IR^m).

(c) Θ induces a solution process Θ⁰ in C_B⁰(Ω, IR^m) provided f = f (t, x, u).

Proof By uniqueness of solutions it suffices to establish induced solution processes in C_B¹(Ω, IR^m) ,→ L^p(Ω, IR^m), and in C_B⁰(Ω, IR^m) ,→ L^p(Ω, IR^m) when f = f (t, x, u). This is done in Lunardi [121], Proposition 7.3.3 for m = 1, and the general case is similar.

Part (c) follows from Theorem 6.2.

Henceforth Θ^k, k ∈ {0, 1} denotes the process Θ⁰ or Θ¹ as in Proposition 6.11.

Dynamics in spaces XΓ

For any set Γ ⊂ IR^m and k = 0, 1 define

X_Γ^k:= {u ∈ C_B^k(Ω, IR^m) : u(Ω) ⊂ Γ}, XΓ := {u ∈ L^p(Ω, IR^m) : u(Ω) ⊂ Γ} (6.12) A rectangle in IR^m is a set of the form J = J1× · · · × Jm where each Ji ⊂ IR is a non-degenerate closed interval. IR^m, IR^m₊ and closed order intervals [a, b], a ≤ b are rectangles.

Note that X_IR^m₊, X_IR^km

+ is the order cone L^p(Ω, IR^m₊), C_B^k(Ω, IR^m)₊ =: C_B^k(Ω, IR^m₊).

Proposition 6.12 Let J := Π^m_i=1Ji be a rectangle in IR^m such that either 0 ∈ Ji or Bi

is Neumann, and the following hold for all x ∈ Ω, u ∈ ∂J:

fi(t, x, u, 0) ≥ 0 if ui = inf Ji, fi(t, x, u, 0) ≤ 0 if ui = sup Ji (6.13) Then:

(i) In the reaction diffusion case, XJ is positively invariant for Θ and X_J^k is positively invariant for Θ^k, (k = 0, 1).

(ii) Suppose k = m = 1 and J ⊂ IR is an interval. Then XJ is positively invariant for Θ and X_J¹ is positively invariant for Θ¹.

Proof For the reaction-diffusion case we sketch a proof that X_J⁰ is Θ⁰-positively invariant using Theorem 6.2. The proof that XJ is Θ-positively invariant follows from this since Θt,t0(u) is the L^p limit limkΘ⁰_t,t0(uk) where uk ∈ X_J⁰ approximates u ∈ XJ in L^p and the facts: Θ⁰ = Θ on X_J⁰, a dense subset of the closed subset XJ. In order to verify the subtangential condition for X_J⁰, it suffices to verify the subtangential condition for J:

lim inf

h&0

1

hdist(u + hf (t, x, u), J) = 0 (6.14) for each (t, x, u) ∈ [0, ∞) × Ω × J by Martin [122] Proposition IX.1.1. But (6.14) is a necessary condition for J to be positively invariant for the ODE

v⁰ = f (t, x, v)

where x is a parameter. See e.g. [122] Theorem VI.2.1. It is well-known and easy to prove that condition (6.13) implies the positive invariance of J for the ODE (see e.g.

Proposition 3.3, Smith and Waltman [204] Proposition B.7, or Walter [226]) Chapter II, sec.12, Theorem II). It follows that (6.14) holds. Therefore the subtangential con-dition for X_J⁰ holds. Finally, we must verify that e^tAX_J⁰ ⊂ X_J⁰ or, equivalently, that e^tAiC_B⁰_i(Ω, Ji) ⊂ C_B⁰_i(Ω, Ji). This follows from Remark 6.7 and standard maximum principle arguments. It also follows from standard comparison principles for parabolic equations. See e.g. Pao [151] Lemma 2.1 or Smith [193] Corollary 2.4.

The case k = m = 1 is a special case of [226] Chapter IV, sec. 25 Theorem II , sec.

31 Corollary V and Corollary IV.

Consider the case that (6.6) is autonomous:

∂ui

∂t = Aiui+ fi(x, u, ∇u), (x ∈ Ω, t > t0) Biui = 0, (x ∈ ∂Ω, t > t0)



, (6.15)

i = 1, . . . , m. The solution processes Θ, Θ¹, Θ⁰ reduce to local semiflows.

We introduce a mild growth condition, trivially satisfied in the reaction-diffusion case:

For each s > 0 there exists C(s) > 0 such that

|v| ≤ s =⇒ |f (x, v, ξ)| ≤ C(s)(1 + |ξ|^2−) (6.16) The following result gives sufficient conditions for solution processes in XΓ to be global, and to admit compact global attractors:

Proposition 6.13 Assume system (6.15) satisfies (6.16). Let Γ ⊂ IR^m be a nonempty compact set such that XΓ is positively invariant for (6.15). Then:

(a) There are solution semiflows Φ, Φ¹ in XΓ, X_Γ¹ respectively. Φ¹ is compact.

(b) Assume (6.15) is reaction-diffusion. Then there is also a solution semiflow Φ⁰ in X_Γ⁰. The semiflows Φ, Φ⁰, Φ¹ are compact and order compact. There is a compact set K ⊂ X_Γ¹ which is the global attractor for all three semiflows.

Proof (a) Let Γ lie in the open ball of radius R > 0 about the origin in IR^m and let h : IR^m → IR^m be any smooth bounded function that agrees with the identity on the open ball of radius R. Define g by g(x, v, ξ) = f (x, h(v), ξ). Every trajectory in XΓ of (6.15) is also a trajectory of the analogous system in which f is replaced by g (compare Pol´aˇcik [159], pages 842-43). Nonlinearity g satisfies (6.16) with C(s) constant so (6.10) holds. As

lim sup

t→τ −

ku(t)k_C⁰_(Ω,IR^m₎ ≤ R,

which implies (6.11), all trajectories are global by Proposition 6.9. Thus, the restrictions of Ψ, Ψ¹ in X and C_B¹(Ω, IR^m) respectively to XΓ and X_Γ¹ define semiflows Φ and Φ¹. As Ψ¹ is compact by Hale [58], Theorem 4.2.2, Φ¹ is compact because X_Γ¹ is closed in C_B¹(Ω, IR^m).

(b) In the reaction-diffusion case a similar argument establishes a compact solution semiflow Φ⁰ in X_Γ⁰; and Φ⁰ is order compact because order intervals in X_Γ⁰ are bounded.

To prove Φ⁰ order compact, let N be an order interval in X_Γ. For every t = 2s > 0, Φs

maps N continuously into an order interval N⁰ of X_Γ⁰. Precompactness in XΓ of ΦtN follows from the precompactness in X_Γ⁰ of Φ⁰_sN⁰,already established, and the continuous inclusion ΦtN = Φ⁰_s◦ ΦsN ⊂ Φ⁰_sN⁰. the same topology) is a compact global attractor for Φ.

We rely on the identity Φ¹_t = Φ⁰_t|X_Γ¹ and continuity of Φ⁰_t : X_Γ⁰ → X_Γ¹ for all t > 0.

As K is invariant under Φ⁰, it follows that K is a compact subset of X_Γ¹. To prove K a global attractor for Φ¹, it suffices to prove: For arbitrary sequences {x(i)} in X_Γ¹, and t(i) → ∞ in IR+ with t(i) >  > 0, there is a sequence ik → ∞ in N such that

Example. Let the ui denote the concentrations or densities of entities such as chem-icals or species. Such quantities are inherently positive, so taking the state space to be L^p(Ω, IR^m₊) or C_B^k(Ω, IR^m₊) is appropriate. We make the plausible assumption that

sufficiently high density levels must decrease. Modeling this situation by (a) and (b) below, we get the following result.

Proposition 6.14 In Equation (6.15) assume f = f (x, u) and let the following hold for i = 1, . . . , m:

(a) fi(x, u) ≥ 0 if ui = 0

(b) there exists κ > 0 such that fi(x, u) < 0 if ui ≥ κ

Then for k = 0, 1 solution processes in the order cones L^p(Ω, IR^m₊), C_B^k(Ω, IR^m₊) are defined by semiflows Φ, Φ^k respectively; and there is a compact set K ⊂ X_[0,κ]^k m that is the global attractor for Φ, Φ⁰ and Φ¹.

Proof Proposition 6.12 and (a) proves L^p(Ω, IR^m₊) and C_B^k(Ω, IR^m₊) are positively invari-ant under the solution process.

Consider the compact rectangles J(c) := [0, cκ]^m ⊂ IR^m, c ≥ 1. Assumption (b) and Proposition 6.12 entail positive invariance of XJ(c). Proposition 6.13 shows that there are solution semiflows in X_J(c) and X_J(c)^k having a compact global attractor Kc⊂ X_J¹_(c) in common. As the J(c) are nested and exhaust IR^m₊, these semiflows come from solution semiflows Φ, Φ^k as required. Moreover, all the attractors Kc coincide with the compact set K := K1 ⊂ X_J(1)¹ . It is easy to see that K is the required global attractor.

Results on global solutions and positively invariant sets can be found in many places.

See for example Amann [9, 10], Cholewa and Dlotko [26], Cosner [33], Lunardi [121], Pol´aˇcik [159], Smith [193], Smoller [181].

Monotone solution processes for parabolic equations

We restrict attention here to monotonicity properties with respect to the standard point-wise and component-point-wise ordering of functions Ω → IR^m: f ≤ g if and only if fi(x) ≤ gi(x) for all x and all i. The natural ordering on L^p(Ω, IR^m) is defined on equivalence classes by the condition on representatives that fi(x) ≤ gi(x) almost everywhere.

Orderings induced by orthants in IRⁿother than the positive orthant can be handled easily by change of variables. See Mincheva [140] and [141] for results in the case of polygonal cones in IRⁿ.

Consider the case m = 1 in Equation (6.8).

Theorem 6.15 In Equation (6.8), assume m = 1 and f is C¹. Then:

(i) Θ is VSOP on L^p(Ω, IR^m).

(ii) Θ¹ is strongly monotone in C_B¹(Ω).

(iii) If f = f (t, x, u) the induced process Θ⁰ on C_B⁰(Ω) is VSOP, and strongly monotone if all boundary operators are Robin.

Proof Let u, v : [t0, t1] × Ω → IR be solutions with v(t0, x) − u(t0, x) ≥ 0 for all x and

where D4f and D3f denote respectively the derivatives of f (t, x, y, ξ) with respect to ξ ∈ IRⁿ and y ∈ IR. By Taylor’s theorem

f (t, x, v, ∇v) − f (t, x, u, ∇u) = b(t, x)(∇u − ∇v) + c(t, x)(u − v), whence (6.17) follows.

The parabolic maximum principle and boundary point lemma ([193], Theorems 7.2.1, 7.2.2) imply that the function w(t1, ·), considered as an element of C_B¹(Ω), is  0. This proves (ii), and the first assertion of (iii) follows from Theorem 6.4 (b). The proof of strong monotonicity for Robin boundary conditions is similar to the arguments given above. Part (i) follows from strong monotonicity of Θ¹, Theorem 6.4 and continuity of Θt,t0 : L^p(Ω, IR^m) → X^α ,→ C_B¹(Ω, IR^m).

For m ≥ 2 we impose further conditions on system (6.6) in order to have a monotone solution process: it must be of reaction-diffusion type, and the vector fields f (t, x, ·) on IR^m must be cooperative. In other words, f (t, x, u) is C¹ in u and ∂fi/∂uj ≥ 0 for all i 6= j. (The latter condition holds vacuously if m = 1). When this holds then the system is called cooperative. If in addition, there exists ¯x ∈ Ω such that the m × m Jacobian matrix [∂fi/∂uj(t, ¯x, u)] is irreducible for all (t, u), we call the system cooperative and irreducible

Theorem 6.16 If system (6.15) is cooperative, then Θ, Θ^k, k = 0, 1 are monotone. If the system is also irreducible, then:

(i) Θ is VSOP on L^p(Ω, IR^m).

(ii) Θ¹ is strongly monotone in C_B¹(Ω, IR^m).

(iii) Θ⁰ is VSOP in C_B⁰(Ω, IR^m) and is strongly monotone when all boundary operators are Robin.

Proof Monotonicity in C_B⁰(Ω, IR^m) follows directly from Theorem 6.5 and Remark 6.6.

Indeed, let u ≤ v in C_B⁰(Ω, IR^m) and t be fixed. Then [F (t, v)−F (t, u)+λ(v−u)](x) =

Z 1 0

∂f

∂u(t, x, su(x) + (1 − s)v(x)) + λI



ds(v−u)(x) ≥ 0

for some λ > 0 and all x ∈ Ω by cooperativity of f and compactness of Ω. This implies that (QM) holds. The positivity of e^tA follows from Lemma 6.10. Monotonicity of Θ in L^p(Ω, IR^m) follows from monotonicity of Θ⁰ and Theorem 6.4.

The proof of VSOP and strong monotonicity for Robin boundary conditions in C_B⁰(Ω, IR^m) is like that of Theorem 6.15(i), exploiting the maximum principle for weakly coupled parabolic systems (Protter and Weinberger [162], Chapter 3, Theorems 13, 14, 15 and page 192, Remark i). See Smith [193], section 7.4 for a similar proof.

Monotonicity of Θ¹ follows from monotonicity of Θ⁰. Strong monotonicity of Θ¹, in the case of Dirichlet boundary conditions, requires exploiting the maximum principle as in the previous case (the same references apply). VSOP of Θ follows from strong monotonicity of Θ¹, Theorem 6.4 and continuity of the composition Θt,t0 : L^p(Ω, IR^m) → X^α → C_B¹(Ω, IR^m).

在文檔中 Monotone Dynamical systems (頁 108-117)