Sharp forced waves of degenerate diffusion equations in shifting environments

1 Introduction

We consider the following degenerate diffusion equation in a shifting environment:

where u ( t , x ) is the population density of the species at location x and time t , and Δ u m with m > 1 represents the porous-medium-type degenerate diffusion, describing density-dependent dispersal due to population pressure. The dependence of resource function r ( ξ ) on the moving variable ξ ≔ x + c t represents the shifting habitat moving at speed c ∈ R . Except for some “trivial” cases, the traveling wave of (1.1) must propagate at the same speed as the shifting environment. Throughout this article, we assume that the resource function r ( ξ ) with ξ = x + c t is continuous and monotonically increasing, r ( ± ∞ ) is finite, and r ( − ∞ ) < 0 < r ( + ∞ ) . Here, the favorable region { x ∈ R ; r ( x + c t ) > 0 } is surrounded by an unfavorable region { x ∈ R ; r ( x + c t ) ≤ 0 } , and we assume that, due to a shifting climate, the patch moves with a fixed speed c > 0 or c < 0 . The environment is getting worse for c < 0 and better for c > 0 .

In recent years, there has been an increasing interest in the study of reaction-diffusion equations with shifting habitats [8,17]. The works in this aspect can be traced back to Berestycki et al. [4], who proposed a mathematical model that involves a reaction-diffusion equation

where u denotes the population density, f denotes the growth function, and c denotes the habitat shifting speed. The existence of forced waves was later extended by Berestycki and Rossi [6,7] to higher-dimensional cases. Subsequently, Berestycki and Fang [5] found that a forced moving Kolmogorov-Petrovski-Piscounov (KPP) nonlinearity f ( x − c t , u ) gives rise to forced traveling waves. They showed the existence, multiplicity, and attractivity of forced traveling waves. The existence and stability of forced waves for reaction diffusion equations in shifting environments with linear diffusion (nondegenerate diffusion) related to (1.2) have been extensively studied [5,12,14,16,20,27]. We also refer the reader to [10,13,17,28] for the case of scalar equations in shifting environments with nonlocal dispersal.

Recently, Liu et al. [18] considered the degenerate diffusion equation (1.1) with m > 1 , and proved the existence of traveling waves for any speed c < 0 (the symbol c therein is opposite to the speed c here) under the condition that r ( ξ ) is monotonically increasing and r ( − ∞ ) < 0 < r ( + ∞ ) . Note that for the classical Fisher-KPP equation with r ( ξ ) ≡ r ( + ∞ ) ≕ r ¯ > 0 and nondegenerate diffusion m = 1 , the traveling waves ϕ ( x + c t ) exist if and only if ∣ c ∣ ≥ c * ( m , r ( + ∞ ) ) = 2 r ¯ in the nondegenerate case m = 1 ; while for degenerate diffusion case with m > 1 , the traveling waves ϕ ( x + c t ) also exist if and only if ∣ c ∣ ≥ c * ( m , r ( + ∞ ) ) , where c * ( m , r ( + ∞ ) ) > 0 is the critical wave speed for the constant environment r ( ξ ) ≡ r ( + ∞ ) , see [1]. Therefore, a shifting nonconstant environment, getting worse and propagating at any fixed speed c < 0 (not necessarily with ∣ c ∣ ≥ c * ( m , r ( + ∞ ) ) ), drags the traveling wave at the same speed, which is called the forced wave.

Since the pioneering work of Aronson [1,2], it has been known that the traveling wave ϕ ( x + c * t ) of degenerate diffusion equations with critical wave speed c * = c * ( m , r ( + ∞ ) ) is of sharp-type, meaning that ϕ ( ξ ) ≡ 0 for all ξ ≤ ξ 0 and ϕ ( ξ ) > 0 for ξ > ξ 0 for some ξ 0 ∈ R . This arises a natural question in the study of degenerate diffusion equations: do forced waves of sharp-type exist? The sharp-type traveling wave with distinct edge of the support reflects the propagation properties of degenerate diffusion equations [3,9,11,24–26], which plays an important role in the study of dynamic behaviors of reaction-diffusion equations with degenerate diffusion. For further studies on porous-medium-type degenerate diffusion, we refer the readers to [2,3,11,19] and the references therein.

The aim of this article is to show the existence of sharp forced waves for degenerate diffusion in shifting environments. We will show that

1. for c < 0 , the forced waves are not sharp;

2. for 0 < c < c * ( m , r ( + ∞ ) ) , there exist forced waves of sharp-type;

3. for c ≥ c * ( m , r ( + ∞ ) ) and additional trivial assumptions, there also exist sharp forced waves with speed c * ( m , r ( + ∞ ) ) , which may not equal to the propagation speed c of the environment.

1. if the environment is getting worse and drags the species in a traveling wave-type, then the wave is not of the sharp-type. Noting that for population dynamics without reproduction, the density decays exponentially and does not vanish in finite time;

2. if the environment is getting better and the species expand to new areas, since the environment near the propagation edge is smaller than r ( + ∞ ) , the propagation speed of the sharp wave, which is equal to the propagation speed of the environment, is smaller than c * ( m , r ( + ∞ ) ) ;

3. if the environment is getting better at a speed greater than or equal to c * ( m , r ( + ∞ ) ) , which is the propagation speed of the species in a constant environment r ( ξ ) ≡ r ( + ∞ ) , then the species cannot catch up with the environment, unless the environment is trivial near + ∞ .

This article is devoted to exploring the role of nonlinear degenerate diffusion and shifting environments in population dynamics: the dependence of sharp-type traveling waves on the propagation speed of the environment. Degenerate diffusion introduces analytical challenges. Here, the important feature of degenerate diffusion equation appears: traveling waves exhibit free boundaries with weak regularity. We prove the existence of a new sharp-type traveling wave with semi-compact support corresponding to the propagation speed of the environment. New existence results of sharp forced waves with weak regularity are established when wave speed c at which the environment is shifting. For this, some new phase transform frameworks are formed to take care of the degeneracy of diffusivity.

The rest of this article is organized as follows: in Section 2, we give the definition of the weak solution and then state the main theorems of this article and in Section 3, we will give the proof of existence of sharp forced waves.

2 Main results

We consider the nonnegative traveling wave solutions, especially the sharp-type traveling waves, of (1.1) with wave speed c ˆ . In most cases, the traveling wave speed c ˆ coincides with the propagation speed c of the habitat environment. We will write c ˆ as c for simplicity, unless they are different (only in Theorem 2.4) under some trivial conditions.

Let u ( t , x ) = ϕ ( ξ ) with ξ = x + c t be a traveling wave-type solution to (1.1), then it satisfies

in some sense. The second-order differential equation (2.1) is degenerate and nonautonomous. The sharp traveling wave may not be smooth, and we present the following definition.

Different from the sharp traveling waves of degenerate diffusion equations in constant environments, the sharp forced waves ϕ ( ξ ) here in Definition 2.1 generally are not translation invariant since the environment r ( ξ ) is heterogeneous. Therefore, the edge ξ 0 is unique and the location of the edge is an important question for the determinacy of the existence of sharp forced traveling waves.

We recall that for constant environments r ( ξ ) ≡ r ( + ∞ ) ≕ r ¯ > 0 , there exists a critical wave speed c * = c * ( m , r ¯ ) > 0 such that the degenerate diffusion equation (1.1) admits traveling waves ϕ ( x + c t ) (monotonically increasing and connecting the positive equilibrium r ¯ at positive infinity) for speed c ≥ c * , and the sharp traveling wave corresponds to the critical wave speed c * , see [1,3] and references therein.

Our main results are stated as follows: when the environment is getting worse, there exists no sharp forced waves.

Theorem 2.2 implies that all the forced waves in [18] are not of sharp-type, i.e., they are positive for all ξ ∈ R and no edge of their supports exists.

The key observation of the article is the following existence of sharp forced traveling wave when the environment is getting better with slow speed.

When the environment is getting better and propagates fast, under some trivial conditions there exist sharp forced traveling waves with speed not necessarily coincides with the environment. In this case c ˆ ≠ c , we write ξ ≔ x + c ˆ t .

Theorem 2.4 is trivial since the sharp traveling wave ϕ ( ξ ) = ϕ ( x + c ˆ t ) is supported in [ ξ 0 , + ∞ ) and propagates to the left in speed c * ( m , r ¯ ) , while the environment is propagating to the left in speed c ≥ c * and the environment r ( x + c t ) is constant for x + c t ≥ ξ ˆ . Note that x + c t ≥ x + c ˆ t ≥ ξ 0 ≥ ξ ˆ at where ξ ≥ ξ 0 , then r ( x + c t ) = r ¯ at those points, meaning that the support of the sharp traveling wave is located within the level set of r ( x + c t ) = r ¯ .

3 Existence of sharp forced waves

In this section, we prove the existence of sharp forced waves when the environment is getting better such that c > 0 . We first focus on the nontrivial case 0 < c < c * ( m , r ¯ ) (Theorem 2.3), followed by the trivial case c ≥ c * ( m , r ¯ ) (Theorem 2.4). The main difficulty of (2.1) lies in the nonautonomous property such that classical phase plane analysis is not applicable. We develop the following generalized phase plane analysis method using phase transform to handle the nonautonomous reaction term.

For any fixed ξ 0 ∈ R , and any sharp profile function ϕ ( ξ ) on ( − ∞ , ξ * ) with ξ * > ξ 0 such that: ϕ ( ξ ) ≡ 0 for all ξ ≤ ξ 0 , 0 < ϕ ( ξ ) < r ¯ for ξ ∈ ( ξ 0 , ξ * ) , ϕ is continuous and monotonically increasing on ( − ∞ , ξ * ) , ( ϕ m ( ξ ) ) ′ ∈ C ( − ∞ , ξ * ) , ϕ ∈ C 2 ( ξ 0 , ξ * ) , ϕ ′ ( ξ ) > 0 for ξ ∈ ( ξ 0 , ξ * ) , and ( ϕ m ( ξ ) ) ′ is locally monotonically increasing near ξ 0 , we define

Note that ϕ ∈ C 2 ( ξ 0 , ξ * ) , then the possible singularity of ( ϕ m ( ξ ) ) ″ must be supported at { ξ 0 } . Since ϕ ( ξ ) = 0 for all ξ ≤ ξ 0 , ( ϕ m ( ξ ) ) ′ ∈ C ( − ∞ , ξ * ) , and ( ϕ m ( ξ ) ) ′ is locally monotonically increasing (thus the variation of ( ϕ m ( ξ ) ) ′ is locally bounded), we see that ( ϕ m ( ξ ) ) ″ ∈ L loc 1 ( − ∞ , ξ * ) , and the above sharp profile function ϕ is a sharp forced wave to (1.1) in the sense of distributions as in Definition 2.1 if and only if ξ * = + ∞ and ϕ ( ξ ) satisfies the differential equation (2.1) in the strong sense (the generalized derivatives ϕ ′ and ( ϕ m ( ξ ) ) ″ are locally integrable) or in the classical sense (the classical derivatives ϕ ′ and ( ϕ m ( ξ ) ) ″ exist) in ( ξ 0 , ξ * ) .

We note that the edge ξ 0 of the sharp forced wave is not a priori known, and for any given ξ 0 ∈ R , there may exist none of the above sharp profile functions that is sharp forced wave to (1.1). Therefore, we first seek for local sharp waves ϕ ( ξ ) such that: the sharp profile ϕ ( ξ ) satisfies the differential equation (2.1) in the classical sense in ( ξ 0 , ξ * ) , where ( − ∞ , ξ * ) with ξ * ∈ ( ξ 0 , + ∞ ] is the maximal existence interval of the sharp profile function ϕ ( ξ ) (especially note that 0 < ϕ ( ξ ) < r ¯ for ξ ∈ ( ξ 0 , ξ * ) ). If for some special ξ 0 such that the local sharp wave exists globally with ξ * = + ∞ , then ϕ ( + ∞ ) = r ¯ and ϕ ( ξ ) is the sharp forced wave with edge ξ 0 , hence the existence of sharp forced waves is proved.

The key step in realizing the above idea is the analysis of local sharp waves for any given ξ 0 (not necessarily the edge of sharp forced wave) and the (continuous and monotone) dependence of local sharp wave on ξ 0 . To achieve this, we translate the nonautonomous and degenerate second-order differential equation (2.1) into a formal first order differential equation with phase transform to handle the nonautonomous term. Noting that the profile function ϕ ′ ( ξ ) > 0 in ( ξ 0 , ξ * ) , the correspondence between ξ ∈ ( ξ 0 , ξ * ) and ϕ ∈ ( 0 , ϕ ( ξ * ) ) is one-to-one, we can regard ξ ∈ ( ξ 0 , ξ * ) as a function of ϕ and take ϕ as an independent variable. In this way, the nonautonomous function r ( ξ ) for ξ ∈ ( ξ 0 , ξ * ) can also be regarded as a function of ϕ , called the phase transform, denoted by

The ordinary differential equation system

corresponding to (2.1) can then be transformed into

If a sharp profile function ϕ ( ξ ) satisfies (3.4) with its phase function ψ being defined in (3.1) and phase transform r ˆ ( ϕ ) being defined in (3.2), then ϕ ( ξ ) is a local sharp wave on ( − ∞ , ξ * ) . The phase transform method dealing with nonautonomous term here can also be employed to handle time-delayed nonlocal term, see [21–23], and convolution-type nonlocal reaction term, see [15], and other generalized phase plane.

We should point out that the phase transform r ˆ ( ϕ ) in (3.2) depends on the specific function ϕ ( ξ ) , while the differential equation (3.4) represents the relation between ψ ( ξ ) = ( ϕ m ( ξ ) ) ′ and ϕ ( ξ ) , which in return gives the specific behavior of ϕ ( ξ ) . Therefore, (3.4) is a differential equation with self-correlation such that r ˆ ( ϕ ) depends on ψ ( ϕ ) in an inexplicit form, representing the nonautonomous property of r ( ξ ) .

We formulate the existence of local sharp wave for any given ξ 0 ∈ R and any given 0 < c < c * ( m , r ¯ ) .

We employ the Schauder’s Fixed Point Theorem to show the local solvability of problem (3.4). Consider the following auxiliary problems:

and

where ( 0 , ϕ i * ) with i = 1 , 2, are the maximal existence intervals. Although the denominator ψ i ( ϕ ) in (3.6) and (3.7) has singularity near ψ i ( 0 ) = 0 , the numerator is a higher-order infinitesimal

Then, solvability of (3.6) and (3.7) with the asymptotic expansion ψ i ( ϕ ) = c ϕ + o ( ϕ ) follows from a basic fixed-point approach.

Let

where C 1 and C 2 are the constants defined in the proof of Lemma 3.2. For any ψ ˜ ∈ X , we can recover a sharp profile function ϕ ˜ ( ξ ) by (3.5) as in Lemma 3.1, corresponding to ψ ˜ ( ϕ ) (though ψ ˜ may not correspond to a local sharp wave), and the phase transform, denoted by r ˜ ( ϕ ) , is defined by (3.2), corresponding to ϕ ˜ ( ξ ) . Let ψ ( ϕ ) be the solution of the following problem:

Note that r ˜ ( ϕ ) in (3.10) depends on ϕ ˜ ( ξ ) , which further depends on the given ψ ˜ ∈ X . The solvability of (3.10) is trivial compared with the self-correlation problem (3.4) and is the same as the solvability of (3.6) and (3.7) proved in Lemma 3.2. Similar to the proof of the monotone dependence Lemma 3.4, we have

that is, ψ ∈ X . Define T : X → X , such that T ( ψ ˜ ) = ψ . Clearly, ‖ ψ ‖ C 1 [ 0 , δ 0 ] are uniformly bounded, T is a compact and continuous operator (the continuity of T is proved in a similar way as the proof of the continuity of T i in Lemma 3.2).

Next, we formulate the monotone dependence of ψ ( ϕ ) on c and ξ 0 . For the sake of convenience, we denote the solution ψ ( ϕ ) ∈ C 1 ( 0 , δ ) of (3.4) with ψ ( 0 ) = 0 , ψ ( ϕ ) > 0 , and ψ ( ϕ ) = c ϕ + o ( ϕ ) for given c and ξ 0 by ψ c , ξ 0 ( ϕ ) . Meanwhile, the local sharp wave ϕ ( ξ ) is denoted by ϕ c , ξ 0 ( ξ ) .