7. SEMILINEAR PARABOLIC EQUATIONS

Stig Larsson

Department of Mathematics Chalmers University of Technology

1996

7.1. Introduction

Many reaction-diffusion problems occurring, for example, in chemistry, biology, ecology, and population genetics, may be modeled by systems of semilinear parabolic equations of the form

(7.1)  \begin{matrix}u_{t}(x, t) − DΔu(x, t) = f(u(x, t), ∇u(x, t)), & x ∈ Ω, 0 < t ≤ T, \\ Bu(x, t) = 0, & x ∈ Γ, 0 < t ≤ T, \\ u(x, 0) = ϕ(x), & x ∈ Ω. \end{matrix}

Here Ω ⊂ R^{d}, d = 1, 2 or 3, is a bounded domain with boundary Γ, u = (u_{1}, …, u_{m}), f : R^{m} → R^{m} is a smooth mapping (the reaction rates), and D = diag (κ_{1}, …, κ_{m}) is a constant, positive definite, diagonal matrix (the diffusion coefficients). The operator B specifies the boundary conditions, which may differ from equation to equation, and which may be of the following types

(7.2)

(Bu)_{i} = κ_{i} \frac{∂u_{i}}{∂n} + α_{i}u_{i},  Robin boundary condition (α_{i} > 0),
(Bu)_{i} = u_{i},  Dirichlet boundary condition (α_{i} = ∞),
(Bu)_{i} = \frac{∂u_{i}}{∂n} ,  Neumann boundary condition (α_{i} = 0),

where n denotes the outward unit normal vector on Γ. See Exercise 7.1 for an example.

Solutions of equations of the form (7.1) may exhibit a variety of phenomena including traveling waves, pattern formation, bifurcations, periodic solutions, ignition, blow-up in finite time, multiple time scales, etc.

Situations where the nonlinearity depends also on x and t, i.e., f = f(x, t, u, ∇u), are also of interest, but will not be considered here for simplicity of presentation. Equations of this form are called semilinear because they are nonlinear only in the lower order derivatives u and ∇u, but linear in the higher order derivatives Δu. Note also that the equations in (7.1) are coupled only through the lower order terms.

In Section 7.2 we formulate the initial boundary value problem (7.1) (with homogeneous Dirichlet boundary conditions) as an ordinary differential equation (dynamical system) in an infinite dimensional phase space. We obtain solutions locally in time under the assumption that the nonlinearity satisfies a suitable Lipschitz condition. The local solutions may be extended globally in time provided that we have an a priori bound. In Section 7.3 we briefly discuss a spatially semidiscrete finite element method within the same framework. Finally in Section 7.4 we study the system (7.1) by means of the maximum principle. We show that it is sometimes possible to obtain a so called invariant set Σ ⊂ R^{m}, such that ϕ(x) ∈ Σ implies u(x, t) ∈ Σ for t ≥ 0. It is then easy to prove existence of global solutions in Σ.

7.2. Semilinear parabolic problems

In this section we first discuss the scalar (m = 1) problem (7.1) with the homogeneous Dirichlet boundary condition. At the end of the section we comment on the general case. We thus consider

(7.3)  \begin{matrix}u_{t}(x, t) − Δu(x, t) = f(u(x, t), ∇u(x, t)), & x ∈ Ω, 0 < t ≤ T, \\ u(x, t) = 0, & x ∈ Γ, 0 < t ≤ T, \\ u(x, 0) = ϕ(x), & x ∈ Ω. \end{matrix}

We begin by recalling some function spaces and norms that will used. Let H = L_{2}(Ω) with the usual norm and inner product

(7.4)  ∥u∥ = (∫_{Ω} |u(x)|^{2} dx)^{1/2}, (u, v) = ∫_{Ω} u(x)v(x) dx,

and let V = H_{0}^{1}(Ω) with norm and inner product

(7.5) ∥u∥_{V} = ∥∇u∥ = (∫_{Ω} |∇u(x)|^{2} dx)^{1/2}, (∇u, ∇v) = ∫_{Ω} ∇u(x) · ∇v(x) dx.

We also introduce the space C([0, T], V) of continuous functions from the interval [0, T] into V. This is a Banach space with the norm

(7.6)  ∥u∥_{L_{∞}([0,T],V)} = sup_{t∈[0,T]} ∥u(t)∥_{V} .

The weak form of (7.3) is

(7.7)  \begin{matrix}u(t) ∈ V for 0 ≤ t ≤ T, u(0) = ϕ, \\ (u_{t}, χ) + (∇u, ∇χ) = (f(u, ∇u), χ), ∀χ ∈ V, 0 < t ≤ T. \end{matrix}

For this to make sense we assume that f induces a nonlinear mapping f̃ : V → H defined by f̃(v)(x) = f(v(x), ∇v(x)), and that this mapping satisfies a local Lipschitz condition: there is an nondecreasing function k such that

(7.8)  ∥f̃(u) − f̃(v)∥ ≤ k(ρ)∥u − v∥_{V} , if ∥u∥_{V} ≤ ρ, ∥v∥_{V} ≤ ρ.

The Lipschitz constant k may thus depend on the size ρ of the functions u and v. If k is independent of ρ then we say that f̃ satisfies a global Lipschitz condition.

We now discuss how the condition (7.8) may be fulfilled in some cases where f = f(u) does not depend on ∇u. The simplest case occurs when f has a bounded derivative:

(7.9)  |f′(s)| ≤ L, ∀s ∈ R.

Then we have

|f(s) − f(t)| = |∫_{0}^{1} f′(θs + (1 − θ)t) dθ (s − t)| ≤ L|s − t|,

so that

(7.10)  ∥f̃(u) − f̃(v)∥ ≤ L∥u − v∥ ≤ CL∥u − v∥_{V} , ∀u, v ∈ V,

where we also employed Poincaré’s inequality ∥u∥ ≤ C∥u∥_{V} for u ∈ V. This means that f̃ satisfies a global Lipschitz condition as a mapping from H into H, and hence also from V into H.

In order to handle more general nonlinearities we need two norm inequalities. We first recall Hölder’s inequality:

(7.11)  ∫_{Ω} |u(x)v(x)| dx ≤ ∥u∥_{L_{q}}∥v∥_{L_{p}}, \frac{1}{q} + \frac{1}{p} = 1,

where

(7.12)  ∥u∥_{L_{p}} = (∫_{Ω} |u(x)|^{p} dx)^{1/p}, 1 ≤ p < ∞; ∥u∥_{L_{∞}} = sup_{x∈Ω} |u(x)|.

The proof of (7.11) is rather elementary and may be found in any book on integration theory. Here we only note that the special case p = q = 2 is Cauchy’s inequality, and that the special case p = 1, q = ∞ is easy:

∫_{Ω} |u(x)v(x)| dx ≤ sup_{x∈Ω} |u(x)| ∫_{Ω} |v(x)| dx = ∥u∥_{L_{∞}}∥v∥_{L_{1}}.

Note that if p is small then q has to be large; Hölder’s inequality thus allows us to estimate the integral ∫ |uv| dx in terms of a weak norm of v if we use a correspondingly stronger norm on u, see Exercise 7.2.

We shall also use the following special case of Sobolev’s inequality:

(7.13)  ∥u∥_{L_{p}} ≤ C∥u∥_{V} , u ∈ V,

which holds for 1 ≤ p ≤ ∞ if d = 1, for 1 ≤ p < ∞ if d = 2, and for 1 ≤ p ≤ 2d/(d−2) if d ≥ 3, see Exercise 7.5. The inequality implies that if the first derivatives of u are square integrable then u itself is integrable to some power p > 2. Note that p is smaller the higher d is. This inequality allows us to control nonlinearities of polynomial growth if we can control the V -norm of the solution. More precisely we have the following lemma.

Lemma 7.1. Assume that f is smooth and, in addition, if d = 2, 3 that f′ satisfies the growth condition

(7.14)  |f′(s)| ≤ C(1 + |s|^{γ}), ∀s ∈ R,

with 1 ≤ γ < ∞ if d = 2, γ = 2 if d = 3. Then the local Lipschitz condition (7.8) is satisfied with k(ρ) = C(1 + ρ^{γ}).

Proof. We first note that

f(u(x)) − f(v(x)) = ω(x)(u(x) − v(x)), with ω(x) = ∫_{0}^{1} f′(θu(x) + (1 − θ)v(x)) dθ.

Hence, employing first Hölder’s inequality (7.11) with q = 3/2, p = 3, and then Sobolev’s inequality (7.13), we get

\begin{aligned}∥f̃(u) − f̃(v)∥ & = (∫_{Ω} |ω(x)|^{2}|u(x) − v(x)|^{2} dx)^{1/2} \\ & ≤ ∥ω∥_{L_{3}}∥u − v∥_{L_{6}} ≤ ∥ω∥_{L_{3}}∥u − v∥_{V} . \end{aligned}

It remains to estimate ω: If d = 2, 3, then

|ω(x)| ≤ ∫_{0}^{1} |f′(θu(x) + (1 − θ)v(x))| dθ
≤ C ∫_{0}^{1}(1 + |θu(x) + (1 − θ)v(x)|^{γ}) dθ [by (7.14)]

≤ C ∫_{0}^{1}(1 + θ|u(x)|^{γ} + (1 − θ)|v(x)|^{γ}) dθ [because h(s) = s^{γ} is convex]

≤ C(1 + |u(x)|^{γ} + |v(x)|^{γ}),

so that, in view of Sobolev’s inequality and the restrictions on γ,

∥ω∥_{L_{3}} ≤ C(∫_{Ω}(1 + |u(x)|^{γ} + |v(x)|^{γ})^{3} dx)^{1/3}
≤ C(∫_{Ω}(1 + |u(x)|^{3γ} + |v(x)|^{3γ}) dx)^{1/3}
= C(1 + ∥u∥_{L_{3γ}}^{γ} + ∥v∥_{L_{3γ}}^{γ} )
≤ C(1 + ∥u∥_{V}^{γ} + ∥v∥_{V}^{γ} ).

Hence ∥u∥_{V} ≤ ρ, ∥v∥_{V} ≤ ρ implies

∥f̃(u) − f̃(v)∥ ≤ C(1 + ∥u∥_{V}^{γ} + ∥v∥_{V}^{γ} )∥u − v∥_{V} ≤ C(1 + ρ^{γ})∥u − v∥_{V} .

If d = 1, then we have ∥v∥_{L_{∞}} ≤ C∥v∥_{V}, so that ∥u∥_{V} ≤ ρ, ∥v∥_{V} ≤ ρ implies ∥ω∥_{L_{∞}} ≤ C(ρ). □

We now return to the initial-boundary value problem (7.3). Defining the operator A = −Δ with domain of definition D(A) = V ∩ H^{2}(Ω) we may write (7.1) as an ordinary differential equation in the infinite dimensional phase space V:

(7.15)  \begin{matrix}u_{t} + Au = f̃(u), 0 < t ≤ T, \\ u(0) = ϕ. \end{matrix}

We say that u is a solution of (7.15) over the interval [0, T] if u ∈ C([0, T], V), and u_{t}(t) ∈ H, u(t) ∈ D(A) for t ∈ (0, T], and if u satisfies (7.15).

Introducing the solution operator E(t) = exp (−tA) of the corresponding linear homogeneous problem, i.e., u(t) = E(t)ϕ is the solution of (7.15) with f̃ = 0, we consider also the nonlinear integral equation

(7.16)  u(t) = E(t)ϕ + ∫_{0}^{t} E(t − s)f̃(u(s)) ds, 0 ≤ t ≤ T.

We say that u is a solution of (7.16) over the interval [0, T] if u ∈ C([0, T], V) and satisfies (7.16). Recall that, by Duhamel’s principle, a solution of (7.15) is also a solution of (7.16), see Exercise 7.6. Conversely, it can be shown that a solution of (7.16) is differentiable so that it is actually a solution of (7.15).

We recall that the operator E(t) satisfies the following bounds:

(7.17)  ∥E(t)ϕ∥_{V} ≤ ∥ϕ∥_{V} ,  t ≥ 0, ϕ ∈ V,
(7.18)  ∥E(t)ϕ∥_{V} ≤ \frac{1}{2}t^{−1/2}∥ϕ∥, t > 0, ϕ ∈ H,

see Exercise 7.3.

After these preparations we may now prove that for any initial value ϕ ∈ V there is a unique solution over a finite time interval [0, T_{0}]; the length of the interval of existence depends on the size of ϕ. The result is therefore called a local existence result. We say that we have global existence if we can prove that solutions exist over an a priori given time interval [0, T] or [0, ∞). See Exercise 7.4 for an example where the solution blows up after a finite time so that global existence fails.

Theorem 7.2. Local existence. For any ρ > 0 there is T_{0} = T_{0}(ρ) such that (7.16), and hence also (7.15), has a unique solution over the time interval [0, T_{0}] for any initial value ϕ ∈ V with ∥ϕ∥_{V} ≤ ρ.

Proof. Let

(7.19)  S(u)(t) = E(t)ϕ + ∫_{0}^{t} E(t − s)f̃(u(s)) ds,

and note that (7.16) is a fixed point equation u = S(u). We shall choose T_{0} and R such that we can apply Banach’s fixed point theorem (the fixed point theorem for contractions) in the closed ball

B = {u ∈ C([0, T_{0}], V) : ∥u∥_{L_{∞}([0,T_{0}],V)} ≤ R}

in the Banach space C([0, T_{0}], V).

We must show (i) that S maps B into itself, (ii) that S is a contraction on B. In order to prove (i) we take u ∈ B and first note that the Lipschitz condition (7.8) implies that

∥f̃(u(t))∥ ≤ ∥f̃(0)∥ + ∥f̃(u(t)) − f̃(0)∥
≤ ∥f̃(0)∥ + k(R)∥u(t)∥_{V}
≤ ∥f̃(0)∥ + k(R)R, 0 ≤ t ≤ T_{0}.

Hence, using also (7.17) and (7.18), we get

∥S(u)(t)∥_{V} ≤ ∥E(t)ϕ∥_{V} + ∫_{0}^{t} ∥E(t − s)f̃(u(s))∥_{V} ds
≤ ∥ϕ∥_{V} + \frac{1}{2} ∫_{0}^{t}(t − s)^{−1/2}∥f̃(u(s))∥ ds
≤ ρ + T_{0}^{1/2}(∥f̃(0)∥ + k(R)R), 0 ≤ t ≤ T_{0}.

This implies

∥S(u)∥_{L_{∞}([0,T_{0}],V)} ≤ ρ + T_{0}^{1/2}(∥f̃(0)∥ + k(R)R).

Choose R ≥ 2ρ and T_{0} = T_{0}(ρ) so small that

(7.20)  T_{0}^{1/2}(∥f̃(0)∥ + k(R)R)≤ \frac{1}{2}R.

Then ∥S(u)∥_{L_{∞}([0,T_{0}],V)} ≤ R and we conclude that S maps B into itself. To show (ii) we take u, v ∈ B and note that

∥f̃(u(t)) − f̃(v(t))∥ ≤ k(R)∥u − v∥_{L_{∞}([0,T_{0}],V)}, 0 ≤ t ≤ T_{0}.

Hence

∥S(u)(t) − S(v(t))∥_{V} ≤ ∫_{0}^{t} ∥E(t − s)(f̃(u(s)) − f̃(v(s)))∥_{V} ds
≤ \frac{1}{2} ∫_{0}^{t}(t − s)^{−1/2}∥f̃(u(s)) − f̃(v(s))∥ ds
≤ T_{0}^{1/2}k(R)∥u − v∥_{L_{∞}([0,T_{0}],V)}, 0 ≤ t ≤ T_{0},

so that

∥S(u) − S(v)∥_{L_{∞}([0,T_{0}],V)} ≤ T_{0}^{1/2}k(R)∥u − v∥_{L_{∞}([0,T_{0}],V)}.

It follows from (7.20) that T_{0}^{1/2}k(R) ≤ \frac{1}{2} and we conclude that S is a contraction on B. Hence S has a unique fixed point u ∈ B. □

Recall that the fixed point is the unique limit of an iteration u_{n} = S(u_{n−1}), which corresponds to a sequence of linear problems

(7.21) \begin{matrix}u_{n,t}(x, t) − Δu_{n}(x, t) = f(u_{n−1}(x, t), ∇u_{n−1}(x, t)), & x ∈ Ω, 0 < t ≤ T, \\ u_{n}(x, t) = 0, & x ∈ Γ, 0 < t ≤ T, \\ u_{n}(x, 0) = ϕ(x), & x ∈ Ω. \end{matrix}

This is a simple example of the “fundamental trick of nonlinear analysis”: linearization.

If f̃ satisfies a global Lipschitz condition as in (7.10), then k is independent of R in the previous proof, so that we may choose T_{0} = (2k)^{−2} and R ≥ 2ρ + ∥f̃(0)∥/k, cf. (7.20). This implies that the time of unique existence T_{0} is independent of ρ, and we can continue the local solution from the interval [0, T_{0}] to the interval [T_{0}, 2T_{0}], and so on. We have global existence.

Corollary. Assume that f̃ : V → H satisfies a global Lipschitz condition. Then (7.15) has a unique solution on any time interval [0, T] for any initial value ϕ ∈ V.

In general we must estimate the local solution a priori over a given time interval [0, T] in order to extend it in this way. An a priori estimate is an estimate of a solution (which is assumed to exist) in terms of the data of the problem; in our case in terms of ϕ and T.

Theorem 7.3. Global existence. Assume that ϕ ∈ V is such that we have an a priori bound

(7.22)  ∥u(t)∥_{V} ≤ ρ, 0 ≤ t ≤ T,

for a possible solution of (7.15). Then (7.15) has a unique solution over the interval [0, T].

Proof. Let T_{0} = T_{0}(ρ) be as in Theorem 7.2. By (7.22) we have ∥ϕ∥_{V} ≤ ρ and hence there is a unique solution on [0, T_{0}]. Now (7.22) implies ∥u(T_{0})∥_{V} ≤ ρ, so that we may solve uniquely over the interval [T_{0}, 2T_{0}] with initial value u(T_{0}). Proceeding like this we continue the solution as long as the a priori bound holds. In a finite number of steps we reach T. □

The possibility for proving an a priori bound usually depends on some special properties of the nonlinearity or of the initial value.

Example. If f(s) = s − s^{3} then |f′(s)| ≤ 3(1 + s^{2}) for all s ∈ R, so that f̃: V → H is locally Lipschitz for d = 1, 2, 3 according to Lemma 7.1. Hence we have local existence for any initial value ϕ ∈ V by Theorem 7.2. In order to obtain global existence we prove an a priori estimate as follows.

With the nonlinear functional

(7.23)  V(v) = ∫_{Ω}(\frac{1}{2}|∇v(x)|^{2} + \frac{1}{4}v(x)^{4} − \frac{1}{2}v(x)^{2}) dx,

and with χ = u_{t} in (7.7) we obtain

∥u_{t}(t)∥^{2} + \frac{d}{dt}V(u(t)) = 0, 0 < t < ∞,

so that

(7.24)  ∫_{0}^{t} ∥u_{t}(s)∥^{2} ds + V(u(t)) ≤ V(ϕ), 0 ≤ t < ∞,

and, in particular,

(7.25)  V(u(t)) ≤ V(ϕ), 0 ≤ t < ∞.

Since 2s ≤ \frac{2^{2}}{2} + \frac{s^{4}}{2} we have \frac{1}{4}s^{4} − \frac{1}{2}s^{2} ≥ \frac{1}{8}s^{4} − \frac{1}{4}, so that

(7.26)  V(u) ≥ \frac{1}{2}∥u∥_{V}^{2} + \frac{1}{8}∥u∥_{L_{4}}^{4} − \frac{1}{4}|Ω| ≥ \frac{1}{2}∥u∥_{V}^{2} − \frac{1}{4}|Ω|,

where |Ω| denotes the volume of Ω. On the other hand, by Sobolev’s inequality (7.13),

V(ϕ) = \frac{1}{2}∥ϕ∥_{V}^{2} + \frac{1}{4}∥ϕ∥_{L_{4}}^{4} − \frac{1}{2}∥ϕ∥^{2} ≤ \frac{1}{2}∥ϕ∥_{V}^{2} + \frac{1}{4}C∥ϕ∥_{V}^{4} .

Hence (7.25) implies the a priori bound

∥u(t)∥_{V}^{2} ≤ \frac{1}{2}|Ω| + ∥ϕ∥_{V}^{2} + \frac{1}{2}C∥ϕ∥_{V}^{4} =: ρ^{2}, 0 ≤ t < ∞.

We conclude that (7.3) with f(s) = s − s^{3} has a unique solution over [0, ∞) for any initial value ϕ ∈ V. Note that the proof of the a priori bound depends on the fact that the leading term in the polynomial f has the right sign, so that the functional V is bounded from below, see (7.26). □

Finally we comment on some generalizations. (i) Theorems 7.2 and 7.3 can obviously be generalized to systems (7.1) with homogeneous boundary conditions Bu = 0 of the form (7.2). The key points are that the phase space V will still be a subspace of (H^{1}(Ω))^{m} and that analogs of (7.17) and (7.18) hold for the corresponding solution operators E(t). (ii) The growth condition in Lemma 7.1 is rather restrictive. However, by replacing ∥ · ∥_{V} by a stronger Sobolev norm we may include smooth nonlinearities f(u, ∇u) with no restriction on the dependence on u and with a certain growth condition with respect to ∇u.

7.3. A spatially semidiscrete finite element approximation

In this section we briefly discuss the approximation of (7.7) with respect to the spatial variables. For simplicity we assume that Ω is a convex polygonal domain. We introduce a family {V_{h}} of finite dimensional subspaces of V. Each space V_{h} consists of those continuous functions that are piecewise linear with respect to a triangulation of Ω with maximum mesh size h, and that satisfy the boundary condition u = 0 on Γ. The spatially semidiscrete finite element problem is

(7.27)  \begin{matrix}u_{h}(t) ∈ V_{h} for 0 ≤ t ≤ T, u_{h}(0) = ϕ_{h}, \\ (u_{h,t}, χ) + (∇u_{h}, ∇χ) = (f(u_{h}, ∇u_{h}), χ), ∀χ ∈ V_{h}, 0 < t ≤ T, \end{matrix}

cf. (7.7). Introducing the discrete Laplacian A_{h} : V_{h} → V_{h}, and the orthogonal projection P_{h} : H → V_{h}, defined by

(A_{h}v_{h}, w_{h}) = (∇v_{h}, ∇w_{h}), (P_{h}v, w_{h}) = (v, w_{h}) ∀v ∈ H, v_{h}, w_{h} ∈ V_{h},

we note that (7.27) may be written as an ordinary differential equation in V_{h}:

(7.28)  \begin{matrix}u_{h,t} + A_{h}u_{h} = P_{h}f̃(u_{h}), 0 < t ≤ T, \\ u_{h}(0) = ϕ_{h}, \end{matrix}

cf. (7.15). Moreover, with the the solution operator E_{h}(t) = exp (−tA_{h}) of the corresponding linear homogeneous problem, i.e., u_{h}(t) = E_{h}(t)ϕ_{h} is the solution of (7.28) with f̃ = 0, we note that (7.28) is equivalent to the nonlinear integral equation

(7.29)  u_{h}(t) = E_{h}(t)ϕ_{h} + ∫_{0}^{t} E_{h}(t − s)P_{h}f̃(u_{h}(s)) ds, 0 ≤ t ≤ T,

cf. (7.16). Since it is easy to prove

(7.30)  ∥E_{h}(t)ϕ_{h}∥_{V} ≤ ∥ϕ_{h}∥_{V} ,  t ≥ 0, ϕ_{h} ∈ V_{h},
(7.31)  ∥E_{h}(t)P_{h}ϕ∥_{V} ≤ \frac{1}{2}t^{−1/2}∥ϕ∥, t > 0, ϕ ∈ H,

see Exercise 7.3, we conclude that Theorems 7.2 and 7.3 and their proofs carry over verbatim to the present situation.

7.4. Invariant sets

We consider first the scalar case (m = 1) of (7.1). We begin by presenting the maximum principle for parabolic problems, more precisely, for functions u that satisfy an inequality of the form

(7.32) u_{t} − Δu + b(x, t) · ∇u + c(x, t)u ≤ 0, with c(x, t) ≥ 0, (x, t) ∈ Ω × (0, T].

We assume that the functions in (7.32) are continuous in \overline{Ω} × [0, T] and “sufficiently smooth” in Ω × (0, T].

Lemma 7.4. If u satisfies (7.32) and c(x, t) > 0 in Ω × (0, T], then u cannot have a positive maximum in Ω × (0, T].

Proof. If u has a positive maximum at a point in Ω × (0, T], then at this point we would have u_{t} ≥ 0, −Δu ≥ 0, ∇u = 0, cu > 0, which contradicts (7.32). □

Note that we assumed that both c and the maximum are positive. This is a preliminary version of the weak maximum principle, which says that if u satisfies (7.32) (with c(x, t) ≥ 0) and the maximum of u over \overline{Ω} × [0, T] is nonnegative then this maximum is attained on the parabolic boundary (\overline{Ω} × \{0\}) ∪ (Γ × [0, T]). In this situation one may draw an even stronger conclusion, namely, that u is constant if it has a nonnegative maximum. This is called the strong maximum principle. Although Lemma 7.4 will be sufficient for our present purpose we state this theorem here without proof for the sake of completeness.

Theorem 7.5. Strong maximum principle. Assume that u satisfies (7.32) and that (x_{0}, t_{0}) ∈ Ω × (0, T]. If

u(x_{0}, t_{0}) = max_{(x,t)∈\overline{Ω}×[0,T]} u(x, t) ≥ 0

then u is constant in \overline{Ω} × [0, t_{0}].

Note that the weak maximum principle follows from the strong maximum principle. Lemma 7.4 is not a special case of the strong maximum principle, because both its assumption and its conclusion are stronger. It is, however, the first step in the proofs of the weak and strong maximum principles.

In order to handle Neumann and Robin boundary conditions we need the following result, which we state without proof.

Lemma 7.6. Boundary point lemma. Assume that u satisfies (7.32) and that (x_{0}, t_{0}) ∈ Γ × (0, T]. If u(x_{0}, t_{0}) ≥ 0 and u(x_{0}, t_{0}) > u(x, t) for all (x, t) ∈ U ∩ (Ω ×

(0, T]), where U is a neighborhood of (x_{0}, t_{0}), then \frac{∂u}{∂n}(x_{0}, t_{0}) > 0.

It is obvious that ∂u/∂n ≥ 0 at a maximum point on Γ. The lemma says that the inequality is strict if (7.32) holds and the maximum is strict and nonnegative.

We now return to nonlinear equations of the form (7.1) with m = 1 and prove one of the main results of this section.

Theorem 7.7. Comparison theorem. Assume that the functions u and v satisfy the inequalities

\begin{matrix}u_{t} − Δu − f(u, ∇u) ≤ v_{t} − Δv − f(v, ∇v), & x ∈ Ω, 0 < t ≤ T, \\ Bu ≤ Bv, & x ∈ Γ, 0 < t ≤ T, \\ u(x, 0) ≤ v(x, 0), & x ∈ Ω. \end{matrix}

Then u(x, t) ≤ v(x, t) for (x, t) ∈ \overline{Ω} × [0, T].

Proof. Let w(x, t) = (u(x, t) − v(x, t))e^{−γt} where the constant γ ≥ 0 is to be chosen later. We must show that w ≤ 0 in \overline{Ω} × [0, T]. It follows from the first of the given inequalities that

w_{t} − Δw + γw − (f(u, ∇u) − f(v, ∇v))e^{−γt} ≤ 0.

Here

f(u, ∇u) − f(v, ∇v) = c(u − v) + b · ∇(u − v),

where

c(x, t) = ∫_{0}^{1} f′_{u}(θu(x, t) + (1 − θ)v(x, t), ∇u(x, t)) dθ,
b(x, t) = ∫_{0}^{1} f′_{∇u}(v(x, t), θ∇u(x, t) + (1 − θ)∇v(x, t)) dθ.

Hence the given inequalities imply

\begin{matrix}w_{t} − Δw − b · ∇w + (γ − c)w ≤ 0, & x ∈ Ω, 0 < t ≤ T, \\ Bw ≤ 0, & x ∈ Γ, 0 < t ≤ T, \\ w(x, 0) ≤ 0, & x ∈ Ω. \end{matrix}

We want to conclude from this that w ≤ 0 in \overline{Ω} × [0, T]. It is already clear that w ≤ 0 on the initial boundary Ω × \{0\}. If w is positive at some point in \overline{Ω} × (0, T], then w has a positive maximum at some (other) point in \overline{Ω} × (0, T]. If we take γ so large that γ − c(x, t) > 0 in Ω × (0, T], then we know from Lemma 7.4 that w cannot have a positive maximum in Ω × (0, T]. Hence w has a positive strict maximum at some point on Γ × (0, T]. At this point we have

Bw = κ\frac{∂w}{∂n} + αw > 0,  (Robin’s boundary condition)
Bw = κ\frac{∂w}{∂n} > 0,  (Neumann’s boundary condition)
Bw = w > 0,  (Dirichlet’s boundary condition)

in view of the boundary point lemma 7.6. This is a contradiction to the inequality Bw ≤ 0. □

By means of the Comparison Theorem it is sometimes possible to construct functions w and W such that

w(x, t) ≤ u(x, t) ≤ W (x, t), (x, t) ∈ Ω × [0, T],

for possible solutions u of (7.1) with m = 1, see Exercises 7.16, 7.17. This is a pointwise a priori bound, i.e., an a priori bound in the maximum norm. Bounds of this type can be used to prove global existence. We do not give the details.

We now generalize this idea to systems (7.1) with m ≥ 1. An invariant set for (7.1) is a subset Σ ⊂ R^{m} such that ϕ(x) ∈ Σ for all x ∈ \overline{Ω} implies u(x, t) ∈ Σ for all (x, t) ∈ \overline{Ω} × [0, ∞). Roughly speaking, one can show that a set Σ is invariant if the vector field f points inwards on the boundary ∂Σ. We shall show this for rectangular sets Σ, i.e., for sets of the form

Σ = ∏_{i=1}^{m}[a_{i}, b_{i}].

We first consider the scalar case.

Lemma 7.8. Let m = 1 and assume that f(a, 0) ≥ 0, f(b, 0) ≤ 0, ϕ(x) ∈ [a, b] for x ∈ \overline{Ω}. For nonhomogeneous Robin or Dirichlet boundary conditions Bu = αg(x, t) assume also that g(x, t) ∈ [a, b] for x ∈ \overline{Ω}, t ≥ 0. Then u(x, t) ∈ [a, b] for x ∈ \overline{Ω}, t ≥ 0.

Proof. We apply the Comparison Theorem 7.7 with v(x, t) = b. If ϕ(x), g(x, t) ∈ [a, b] for x ∈ \overline{Ω}, then

u_{t} − Δu − f(u, ∇u) = 0 ≤ −f(b, 0) = v_{t} − Δv − f(v, ∇v),  x ∈ Ω, t > 0,
Bu = κ\frac{∂u}{∂n} + αu = αg ≤ αb = κ\frac{∂b}{∂n} + αb = Bv, (Robin/Dirichlet) x ∈ Γ, t > 0,
Bu = κ\frac{∂u}{∂n} = 0 = κ\frac{∂b}{∂n} = Bv, (Neumann)  x ∈ Γ, t > 0,
u(x, 0) = ϕ(x) ≤ b = v(x, 0),  x ∈ Ω.

We conclude that u(x, t) ≤ v(x, t) = b. The inequality u(x, t) ≥ a is proved in the same manner. □

Thus the result applies to the case of nonhomogeneous Robin/Dirichlet and homogeneous Neumann boundary conditions. We assume that f points inwards on the boundary of [a, b] and that the data ϕ, g belong to [a, b] (for the homogeneous Neumann boundary condition there is no g). It is now clear how to generalize this result to the case of a system.

Theorem 7.9. Let Σ = ∏_{i=1}^{m}[a_{i}, b_{i}] and assume that ϕ(x), g(x, t) ∈ Σ and that

f_{i}(v, 0) ≥ 0, if v_{i} = a_{i}, v_{j} ∈ [a_{j}, b_{j}], j ≠ i,
f_{i}(v, 0) ≤ 0, if v_{i} = b_{i}, v_{j} ∈ [a_{j}, b_{j}], j ≠ i.

Then u(x, t) ∈ [a, b] for x ∈ \overline{Ω}, t ≥ 0.

Exercises

7.1. Combustion. Consider an exothermic chemical reaction taking place in a fluid diffusing through a porous medium, e.g., a chemical reaction in a catalyst pellet, or self-ignited combustion in a pile of sawdust. Using the Arrhenius law for the temperature dependence of the reaction rate, the balance of mass and heat are modeled by the coupled parabolic equations

ϵ\frac{∂c}{∂t̂} − DΔc = −Ac^{p} exp (−E/(RT)),
C_{P} \frac{∂T}{∂t̂} − kΔT = HAc^{p} exp (−E/(RT)),

in the spatial domain Ω̂. Here c denotes the concentration of the reactant diffusing in the porous medium and T is the temperature. The quantities ϵ (the porosity), C_{P} (the heat capacity), D (the diffusion coefficient), k (the thermal conductivity), A (a rate constant), H (the heat of reaction), E (the activation energy), R (the universal gas constant) and the integer p (the order of the reaction) are assumed to be constant. The domain is assumed to be surrounded by fluid containing a fixed concentration c_{f} of the reactant and which is maintained at a constant temperature T_{f}. The boundary conditions are then

D \frac{∂c}{∂n} + ϵh_{c}(c − c_{f}) = 0, k \frac{∂T}{∂n} + h_{T} (T − T_{f}) = 0,

where h_{c}, h_{T} are transfer coefficients for mass and heat. Show that these equations can be put in dimensionless form

(7.33)

Lv_{t} − Δv = −Φ^{2}(1 + v)^{p} exp(u/(1 + u/γ)),
u_{t} − Δu = βγΦ^{2}(1 + v)^{p} exp(u/(1 + u/γ)),

x ∈ Ω, t > 0,

\frac{∂v}{∂n} + νv = 0, \frac{∂u}{∂n} + μu = 0,  x ∈ Γ, t > 0,
v(x, 0) = v_{0}(x), u(x, 0) = u_{0}(x),  x ∈ Ω.

Here v = (c − c_{f})/c_{f} and u = γ(T − T_{f})/T_{f} denote the dimensionless concentration and temperature, and the coefficients are γ = E/(RT_{f}) (the Arrhenius number), Φ (the Thiele modulus), β (the Prater temperature), L (the Lewis number), and ν, μ (the Biot numbers for mass and heat). Show that this problem can be put in the framework of Section 7.2 if p is not too large. Use the maximum principle to show that −1 ≤ v(x, t) ≤ 0, u(x, t) ≥ 0 if −1 ≤ v_{0}(x) ≤ 0, u_{0}(x) ≥ 0.

7.2. Show that ∥u∥_{L_{p}} ≤ C∥u∥_{L_{q}}, C = |Ω|^{1/p−1/q}. This means that ∥ · ∥_{L_{q}} is a stronger norm than ∥ · ∥_{L_{p}}, if q ≥ p. Hint: apply Hölder’s inequality to the integral ∫_{Ω} 1 · |u(x)|^{p} dx.

7.3. Prove (7.17)–(7.18) and (7.30)–(7.31). Hint: estimate the solutions of (7.7) and (7.27) with f = 0 by means of the energy method or the spectral method.

7.4. Blow-up in finite time. Consider the problem

\begin{matrix}u_{t} − Δu = u^{3}, & x ∈ Ω, t > 0, \\ u(x, t) = 0, & x ∈ Γ, t > 0, \\ u(x, 0) = ϕ(x), & x ∈ Ω. \end{matrix}

(a) Show that this fits in the framework of Section 7.2 so that local solutions exist for all ϕ ∈ V.

(b) With V(v) = \frac{1}{2}∥∇v∥^{2} − \frac{1}{4}∥v∥_{L_{4}}^{4} show that V(u(t)) ≤ V(ϕ).

(c) Show that V(ϕ) ≤ 0 if ∥ϕ∥ is sufficiently large. Hint: write ϕ = αϕ̂ with α ≥ 0, ∥ϕ̂∥ = 1, use the inequality ∥v∥^{4} ≤ |Ω| ∥v∥_{L_{4}}^{4} from Exercise 7.2.

(d) Show that if V(ϕ) ≤ 0, then

∥u(t)∥^{2} ≥ \frac{∥ϕ∥^{2}}{1 − t|Ω|^{−1}∥ϕ∥^{2}} , 0 ≤ t < T_{1} = \frac{|Ω|}{∥ϕ∥^{−2}} .

Thus the maximal time of existence T_{max} ≤ T_{1} < ∞ if V(ϕ) ≤ 0. Hint: show

\frac{d}{dt}∥u(t)∥^{2} = −4V(u(t)) + ∥u(t)∥_{L_{4}}^{4} ≥ −4V(ϕ) + ∥u(t)∥_{L_{4}}^{4},

conclude \frac{d}{dt}∥u(t)∥^{2} ≥ |Ω|^{−1}∥u(t)∥^{4}, and solve this inequality.

(e) Assume that T_{max} < ∞. From Exercise 7.14 we know that ∥u(t)∥_{V} → ∞ as t → T_{max}. Show that ∥u(t)∥_{L_{4}} → ∞ as t → T_{max}. Hint: use the result from part (b).

7.5. Prove Sobolev’s inequality (7.13) for Ω = (0, 1), d = 1, and for Ω = (0, 1)×(0, 1), d = 2. The proof is a bit more complicated for more general domains, see for example [?].

7.6. Prove that (7.15) implies (7.16). Hint: if u_{s}(s) ∈ H, u(s) ∈ D (A) for 0 < s <

t ≤ T then \frac{d}{ds}(E(t − s)u(s)) = E(t − s)f̃(u(s)).

7.7. Gronwall’s lemma. Assume that φ, a, b are non-negative functions, that a is nondecreasing (a(t) ≥ a(s) if t ≥ s), and that

φ(t) ≤ a(t) + ∫_{0}^{t} b(s)φ(s) ds, 0 ≤ t ≤ T.

Then

φ(t) ≤ a(t) exp(∫_{0}^{t} b(s) ds), 0 ≤ t ≤ T.

Hint: set Φ(t) = ∫_{0}^{t} b(s)φ(s) ds, show that Φ′(t) ≤ b(t)(a(t) + Φ(t)), and apply an integrating factor.

7.8. Burgers’ equation. (a) Show that the problem

u_{t} − u_{xx} + uu_{x} = g(x), x ∈ (0, 1), t > 0,
u(0, t) = u(1, t) = 0,  t > 0,
u(x, 0) = ϕ(x),  x ∈ (0, 1),

fits in the framework of Section 7.2 so that local solutions exist. Hint: ∥v∥_{L_{∞}} ≤ ∥v∥_{V}. (b) Prove the a priori bounds

∥u(t)∥^{2} + ∫_{0}^{t} ∥u(s)∥_{V}^{2} ds ≤ ∥ϕ∥^{2} + t∥g∥^{2},
∥u(t)∥_{V}^{2} ≤ (∥ϕ∥_{V}^{2} + t∥g∥^{2})exp(∥ϕ∥^{2} + t∥g∥^{2}).

Conclude that ∥u(t)∥_{V} ≤ C for 0 ≤ t ≤ T, where C depends on the data ϕ, g, T, and hence that there is a unique solution over any time interval [0, T]. Hint: for the second estimate show first

∥u(t)∥_{V}^{2} ≤ ∥ϕ∥_{V}^{2} + t∥g∥^{2} + ∫_{0}^{t} ∥u(s)∥_{V}^{4} ds,

and apply Gronwall’s lemma with φ(t) = ∥u(t)∥_{V}^{2} , b(t) = ∥u(t)∥_{V}^{2} .

7.9. Prove Theorem 7.2 by means of the energy method without using (7.16). More precisely, define the nonlinear operator u = S(v) for v ∈ C([0, T], V) as the solution of (7.7) with f(u(t)) replaced by f(v(t)). Show by an energy estimate that S is a contraction on a suitable closed set in the Banach space C([0, T], V) ∩ H^{1}([0, T], H) with norm ∥v∥_{L_{∞}([0,T],V)} + ∥v_{t}∥_{L_{2}([0,T],H)}.

7.10. Generalized Gronwall lemma. Assume that the nonnegative function φ satisfies the inequality

φ(t) ≤ a + b ∫_{0}^{t}(t − s)^{−1/2}φ(s) ds, 0 ≤ t ≤ T,

where a ≥ 0, b ≥ 0 are constants. Then

φ(t) ≤ a(1 + 2bt^{1/2})e^{Cb^{2}t}, 0 ≤ t ≤ T,

where C = ∫_{0}^{1}(1 − s)^{−1/2}s^{−1/2} ds. Hint: iterate the given inequality once, then apply the standard Gronwall lemma.

7.11. Continuous dependence on initial data. Let T : V → C([0, T_{0}], V) denote the local solution operator defined in Theorem 7.2 by the relation u(t) = (T (ϕ))(t). Show that T satisfies a local Lipschitz condition. Hint: you will need the generalized Gronwall lemma of Exercise 7.10.

7.12. An error bound. Assume that u, u_{h} are solutions of (7.15) and (7.28), respectively, with ϕ_{h} = P_{h}ϕ, and ∥u(t)∥_{V} ≤ R, ∥u_{h}(t)∥_{V} ≤ R for 0 ≤ t ≤ T. Show that

∥u_{h}(t) − u(t)∥_{V} ≤ C(R, T)h sup_{0<s≤T}(∥u(s)∥_{H^{2}} + s^{1/2}∥u_{t}(s)∥_{H^{1}}), 0 ≤ t ≤ T.

7.13. Ljapunov functional. Let V : V → R be a functional. We say that V is a Ljapunov functional for problem (7.15) if the following conditions are satisfied:

(1) V is continuous;

(2) V is bounded from below;

(3) V(v) → ∞ as ∥v∥_{V} → ∞;

(4) V(u(t)) is nondecreasing with respect to t for any solution u of (7.15);

(5) if a solution u of (7.15) satisfies V(u(t)) = V(ϕ) for all t ≥ 0 then u is a

stationary solution. Show that the functional defined in (7.23) is a Ljapunov functional for the problem described in the Example.

7.14. The maximal interval of existence. Assume that f̃ is such that the local existence theorem 7.2 can be applied. Let T_{max} = T_{max}(ϕ) be the maximal time of existence of a solution u of (7.15) with initial value ϕ. Show the following variant of the global existence theorem 7.3: if T_{max} < ∞, then there is a sequence t_{n} such that t_{n} < T_{max}, t_{n} → T_{max} and ∥u(t_{n})∥_{V} → ∞ as n → ∞. Thus, an a priori bound ∥u(t)∥_{V} ≤ C for 0 ≤ t < ∞ implies T_{max} = ∞, i.e., global existence. Hint: assume that the conclusion is false and apply Theorem 7.2. By an additional argument one may show that actually ∥u(t)∥_{V} → ∞ as t → T_{max}, if T_{max} < ∞. This means that the whole “curve” ∥u(t)∥_{V} blows up, not just a sequence of points ∥u(t_{n})∥_{V}.

7.15. Prove the comparison theorem 7.7 by referring to the strong maximum principle 7.5 instead of Lemma 7.4.

7.16. Consider

\begin{matrix}u_{t} − Δu = −u^{3}, & x ∈ Ω, t > 0, \\ \frac{∂u}{∂n} = 0, & x ∈ Γ, t > 0, \\ u(x, 0) = ϕ(x), & x ∈ Ω. \end{matrix}

Let m ≤ ϕ(x) ≤ M for x ∈ Ω and show that w(t) ≤ u(x, t) ≤ W (t) for x ∈ Ω, t ≥ 0, where w, W are the solutions of the ordinary differential equations

w′(t) = −w(t)^{3}, w(0) = m; W′(t) = −W (t)^{3}, W (0) = M.

Conclude that u(x, t) → 0 as t → ∞.

7.17. Consider again the combustion problem (7.33), to be specific, with the Neumann and Dirichlet boundary conditions. More precisely, consider

(7.34)  \begin{matrix}\begin{matrix}v_{t} − Δv = −αvf(u), \\ u_{t} − Δu = βγαvf(u), \end{matrix} & x ∈ Ω, t > 0, \\ \frac{∂v}{∂n} = 0, u = 0, & x ∈ Γ, t > 0, \\ v(x, 0) = v_{0}(x), u(x, 0) = u_{0}(x), & x ∈ Ω. \end{matrix}

were α, β, γ > 0, f(s) = exp(s/(1 + s/γ)), and where v = c/c_{f}, u = γ(T − T_{f})/T_{f}. Assume 0 ≤ v_{0}(x) ≤ 1, u_{0}(x) ≥ 0 for x ∈ Ω. Show that

0 ≤ v(x, t) ≤ e^{−αt}, 0 ≤ u(x, t) ≤ w(x, t), (x, t) ∈ \overline{Ω} × [0, ∞),

where w is the unique solution of

\begin{matrix}w_{t} − Δw = βγαe^{γ−αt}, & x ∈ Ω, t > 0, \\ w = 0, & x ∈ Γ, t > 0, \\ w(x, 0) = u_{0}(x), & x ∈ Ω. \end{matrix}

Conclude that v(x, t) → 0 and u(x, t) → 0 as t → ∞. If γ is large then the temperature u may become very large before it finally decays to 0. This possibility is related to the possibility of self-ignition in the combustion problem. Hint: 1 ≤ f(s) ≤ e^{γ} for s ≥ 0.

7.18. The Frank-Kamenetsky equation. When γ is large in the combustion problem (7.34) one can show that v changes more slowly than u. Assuming that v is constant, and introducing the approximation f(s) = exp(s/(1 + s/γ)) ≈ exp(s) we now consider

(7.35)  \begin{matrix}u_{t} − Δu = λe^{u}, & x ∈ Ω, t > 0, \\ u = 0, & x ∈ Γ, t > 0, \\ u(x, 0) = 0, & x ∈ Ω, \end{matrix}

where λ > 0, and where we have taken the initial value u_{0}(x) ≡ 0.

(a) Show that solutions of (7.35) are positive and nondecreasing with respect to time, i.e., u(x, t_{1}) ≤ u(x, t_{2}) for t_{1} ≤ t_{2}. Hint: differentiate the equation (7.35) with respect to t to obtain a differential equation for w = u_{t} and apply the maximum principle.

(b) Assume that the associated stationary problem

(7.36)  \begin{aligned}−Δv & = λe^{v}, & x ∈ Ω, \\ v & = 0, & x ∈ Γ, \end{aligned}

has a solution v. (i) Show that v(x) > 0. Hint: Note that λe^{v} > 0 and argue as in the proof of Lemma 7.4. (ii) Show that (7.35) has a solution over the time interval [0, ∞). Hint: 0 ≤ u(x, t) ≤ v(x). It can be shown that u(x, t) converges monotonically to the smallest solution of (7.36) as t → ∞.

(c) Show that (7.36) has at least one solution when λ is sufficiently small, and no solution when λ is sufficiently large. Hint: In order to show that there is no solution for large λ, let φ be an eigenfunction of the eigenvalue problem

(7.37)  \begin{aligned}−Δφ & = μφ, & x ∈ Ω, \\ φ & = 0, & x ∈ Γ, \end{aligned}

associated with its smallest eigenvalue μ. It can be shown that this eigenfunction does not change sign so that we may assume that φ(x) > 0. Form the inner product of (7.36) with φ and the inner product of (7.37) with v, and note that e^{s} ≥ es for all s. Use the contraction mapping theorem to show that there is at least one solution for small λ.

(d) Show that ∥u(·, t)∥_{L_{∞}} does not remain bounded when λ is sufficiently large. Hint: note that (u, φ) ≤ ∥u∥_{L_{∞}}∥φ∥_{L_{1}}, and derive a differential inequality for (u, φ). Solve this inequality starting from some time t_{0} > 0 using the fact that u(x, t_{0}) > 0 according to (a).

(e) These results indicate that thermal self-ignition occurs when λ is large but not when λ is small.

7.19. The FitzHugh-Nagumo equations. The following equations are a simple model for the conduction of potentials along nerve fibres:

\begin{matrix}\begin{matrix}u_{t} − u_{xx} = f(u) − v, \\ v_{t} − δv_{xx} = σu − γv, \end{matrix} & x ∈ (0, 1), t > 0, \\ u(0, t) = u(1, t) = 0, v(0, t) = v(1, t) = 0, & t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ (0, 1), \end{matrix}

where f(s) = s(1−s)(s−a), and a, δ, σ, γ are positive constants with 0 < a < \frac{1}{2}. Find an invariant set. Hint: sketch the zero curves of the source terms in the uv-plane and determine the regions where they are positive and negative.

7.20. The Volterra-Lotka model for population dynamics. Consider two populations that are distributed in a spatial domain Ω with densities u, v. Allowing for random motion of the individuals we have

\begin{matrix}\begin{matrix}u_{t} − d_{1}Δu = uM(u, v), \\ v_{t} − d_{2}Δv = vN(u, v), \end{matrix} & x ∈ Ω, t > 0, \\ \frac{∂u}{∂n} = 0, \frac{∂v}{∂n} = 0, & x ∈ Γ, t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ Ω, \end{matrix}

where d_{1}, d_{2} are positive constants.

(a) Competing species. For the situation where two species are competing for a common ecological niche we assume that

M′_{v} < 0, N′_{u} < 0.

More specifically, take

M(u, v) = A_{1} − B_{1}u − C_{1}v,
N(u, v) = A_{2} − B_{2}u − C_{2}v,

where A_{i}, B_{i}, C_{i} are positive constants (A_{i} represent the natural rates of growth, the other terms describe competition of the members of one species among themselves and with members of the other species). Find invariant sets. Note that depending on A_{i}, B_{i}, C_{i} we may have three or four stationary solutions, and several invariants sets are possible.

(b) The predator-prey equations. For predators v feeding on prey u we assume that

M′_{v} < 0, N′_{u} > 0.

More specifically, take

M(u, v) = (u − δ)(1 − u) − v,
N(u, v) = −μ − αv + u,

where 0 < δ < 1, α > 0, δ < μ < 1. Find invariant sets.

(c) Symbiosis. In this case we have M′_{v} > 0, N′_{u} > 0.