Curved fronts of bistable reaction–diffusion equations with nonlinear convection

where f is the nonlinear reaction term and (g(u))y is the nonlinear convection term. In general, the term (g(u))y represents a convective or advective phenomenon, with g ′(u) denoting a nonlinear velocity function. As a matter of fact, reaction–diffusion equations with convection term are widely used to model some reaction–diffusion processes taking place in moving media such as fluids, for example, combustion, atmospheric chemistry, and plankton distributions in the sea, see Berestycki [1], Cencini et al. [6], Gilding and Kersner [21], Murray [41], and the references therein. Of particular interest is the influence of advection terms on the propagation of traveling wave fronts, which were studied by many researchers, see Berestycki [1], Crooks [8–10], Crooks and Mascia [11], Crooks and Toland [12], Crooks and Tsai [13], Gilding [20], Gilding and Kersner [21], Malaguti and Marcelli [36, 37], Malaguti et al. [38], Volpert et al. [52]. In this paper we assume that f ∈ C2(R) satisfies the following conditions: (F) (i) f (0) = f (1) = 0, f ′(0) < 0, f ′(1) < 0; (ii) {r ∈ [0, 1] : f (r) = 0} = {0,λ, 1} with f ′(λ) > 0; (iii) ∫ 1 0 f (r) dr > 0; (iv) f (r) < 0, f ′(r) < 0 for r > 1; f (r) > 0, f ′(r) < 0 for r < 0.


Introduction
In this paper, we consider traveling wave solutions of the following reaction-diffusion equations with a nonlinear convection term: u t + g(u) y = u + f (u), (x, y) ∈ R 2 , t > 0, (1.1) where f is the nonlinear reaction term and (g(u)) y is the nonlinear convection term. In general, the term (g(u)) y represents a convective or advective phenomenon, with g (u) denoting a nonlinear velocity function. As a matter of fact, reaction-diffusion equations with convection term are widely used to model some reaction-diffusion processes taking place in moving media such as fluids, for example, combustion, atmospheric chemistry, and plankton distributions in the sea, see Berestycki [1], Cencini et al. [6], Gilding and Kersner [21], Murray [41], and the references therein. Of particular interest is the influence of advection terms on the propagation of traveling wave fronts, which were studied by many researchers, see Berestycki [1], Crooks [8][9][10], Crooks and Mascia [11], Crooks and Toland [12], Crooks and Tsai [13], Gilding [20], Gilding and Kersner [21], Malaguti and Marcelli [36,37], Malaguti et al. [38], Volpert et al. [52].
For each θ ∈ [0, 2π], a planar traveling front of (1.1) with direction θ means a function u(x, y, t) = U θ (X), X = x cos θ + y sin θ + c θ t satisfying where c θ ∈ R is called the wave speed. It is obvious that the existence of the solution pair (U θ , c θ ) satisfying (1.2) is equivalent to the existence of traveling wave fronts of the following equation in a one-dimensional space: which has been extensively studied. In 1998, Crooks and Toland [12] considered traveling wave fronts of the more general reaction-diffusion-convection system where D is a positive-definite diagonal matrix, F : R N → R N is continuously differentiable and is of bistable type, G is a continuously differentiable, diagonal-matrix-valued function on R N × R N , and there exist continuous functions β, γ : [0, ∞) → [0, ∞) such that, for each u, v ∈ R N , G satisfies where β is increasing and β(p)/p → 0 as p → ∞. They showed that there exists a unique speed c for which (1.3) has an increasing traveling front φ satisfying Dφ + cφ + G φ, φ φ + F(φ) = 0 and connecting two stable equilibria of (1.3). Furthermore, Crooks [8] showed the global stability of traveling front φ if the initial-value u 0 (x) is bounded, uniformly continuously differentiable and such that φ(x)u 0 (x) is small when |x| is large. Later, Crooks [9] studied the existence and stability of traveling-front solutions for the following gradient-dependent system: where D is a positive-definite diagonal matrix and f is a "monostable" function. Crooks [9] showed that if f satisfies some given conditions, then there exists a critical wave speed c * ∈ R such that there exists a monotone traveling front solution if and only if c ≥ c * . Furthermore, the stability of traveling front solutions for system (1.4) was proved. It should be emphasized that a special interest is to consider the case that the diffusion coefficient D of (1.3) and (1.4) is vanished. In 1997, Mascia [39] established the existence of entropy traveling fronts for the balance law where g is a convex function while f is bistable or monostable. In 2000, Mascia [40] proved the existence of entropy traveling front solutions for (1.5) with nonconvex flux g and monostable reaction f , that is, the flux g is assumed to be smooth and is allowed to have finitely many points of inflection. Thanks to Crooks [9] and Mascia [39,40], Crooks and Mascia [11] considered the convergence as ε → 0 of traveling front speeds for the parabolic equation (1.6) to front speeds for the balance law (1.5). They assumed that the flux g is smooth and may have points of inflection and the reaction term f is of monostable type, with simple zeroes at 0 and 1 and negative in between. They proved that the minimal speed c * of fronts for (1.5) defined by using entropy criteria coincides with the vanishing-diffusion limit of the minimal speeds c * ε for (1.6). Afterwards, Crooks [10] established the L 1 (R)-convergence of corresponding traveling-front profiles w ε with speed c ε (minimal or non-minimal speed) and w ε (0) = 1/2 for (1.6) in the limit ε → 0. Namely, as ε → 0, where w is the profile of the unique entropy traveling-front solution of (1.5) with speed c (minimal or non-minimal speed) and w(0) = 1/2. More recently, Crooks and Tsai [13] established the existence and uniqueness of entire solutions for both monostable and bistable nonlinearity. Especially, they also considered the case that ε → 0.
Here we would like to mention that the main method of this paper comes from Ninomiya and Taniguchi [43] and Wang [54]. Nevertheless, to the best of our knowledge, this paper is the first to consider traveling curved fronts for a reaction-diffusion equation with nonlinear convection in R 2 . This paper is organized as follows: In Sect. 2, we prove the existence of the traveling curved front v by constructing an appropriate supersolution of (1.10). In Sect. 3, we show the asymptotic stability of the traveling curved front v, namely, we prove (1.14).
In the remainder of this paper we always assume that (F) and (G) hold and θ ∈ (0, π 2 ) satisfies assumption (C). Moreover, we also assume that c θ > 0. Let (U θ (·), c θ ) be defined by (1.2), and let s θ := c θ sin θ > 0. In this case, we also have For the sake of convenience, in the sequel we always denote (U θ (·), c θ ) and s θ by (U(·), c) and s, respectively.

Existence
In this section we show the existence of traveling curved fronts of (1.1). It follows from Ninomiya and Taniguchi [43] that there exists a unique function ϕ(x) with asymptotic lines y = m * |x| satisfying The readers can refer to Fig. 3 in Ninomiya and Taniguchi [43] for the shape of the function ϕ. It follows from Ninomiya and Taniguchi [43, Lemma 2.1] that there exist positive constants β 2 := sm * , C j (j = 2, 3, 4) and ν ± such that for all x ∈ R, where M * is a bounded positive constant and We note that From (2.1) and (2.2), one observes that Since U(X) is increasing in X ∈ R, we define that positive constants A and B are large enough satisfying respectively. Then, if we have that -A ≤ X ≤ B. Furthermore, it follows from assumption (G) that there exist positive constants l 1 and l 2 such that Now, we give the definitions of supersolution and subsolution of (1.9).
Similarly, we can define a subsolution u(x, z, t) by reversing the inequality in (2.7).
The next lemma gives a supersolution of (1.9).
In the following, we give the existence of traveling curved fronts of (1.1).

Theorem 2.3
There exists a traveling wave solution u(x, y, t) = v * (x, y + st) of (1.1) satisfying (1.10) and Proof To establish a traveling curved front of (1.1), we first construct a classical solution v * of the stationary equation (1.10). Let where C 1 and l 2 are as in (1.8) and (2.6), respectively. Consider the following linear initial value problem: Theorem 5.1.4 (iv)], there exists a constant C > 0 such that where ∂ ∂n is the outward normal derivative on ∂Ω.
Then we have proved that u(·, ·, t; v -) is monotone increasing with respect to t.
Let us show that u(x, z, t; v -) is monotone increasing with respect to z. Taking the derivative of equation (2.20) with respect to z, we have Using the comparison principle [14, Theorem 25.6], we have u z (x, z, t; v -) > 0. As above, we conclude that the limit lim t→∞ u(x, z, t; v -) := u 1 (x, z) exists. It follows from (2.21) that u 1 (x, z) ∈ C 2+α (R 2 ) and Now we show that u 1 further satisfies For T > 0, multiplying both sides of the aforementioned equality by 1 T and integrating over (T, 2T), we obtain Due to the arbitrariness of φ ∈ C ∞ 0 (R 2 ), we conclude that equality (2.24) holds. By virtue of assumption (G) and the definition of N , we have By (2.24) and (2.25), we know that u 1 (x, z) is a subsolution of the following problem: The local existence of a unique solution w(x, z, t; u 1 ) of the last equation follows from [ By the arguments similar to those for u(x, z, t; v -) and u 1 (x, z), we have that w(x, z, t; u 1 ) is monotone increasing in t > 0 and the limit function v * (x, z) := lim t→∞ w x, z, t; u 1 (2.26) exists. In particular, v * (x, z) satisfies v * (·) C 2+α (R 2 ) ≤ C with some constant C > 0 and Since ε ∈ (0, ε + 0 ) and α ∈ (0, α + 0 ) are arbitrary, it follows from (2.8) that In addition, it is clear that v * (x, z) < 1 for any (x, z) ∈ R 2 . This completes the proof.

Global asymptotic stability
In this section we develop the arguments of Ninomiya and Taniguchi [43] to establish the stability of traveling curved front v * obtained in Sect. 2. We prove that (1.14) holds true for u 0 (x, z) ≥ v -(x, z). See Theorem 3.6. Consider the following initial value problem: where u 0 ∈ BUC 1 (R 2 ) is a given initial function. The global existence of a unique solution w(x, z, t; u 0 ) of equation ( Using [34, Chapter VII, Theorem 5.1], we further have that there exists K (u 0 ) > 0 such that w(·, t; u 0 ) C 2 (R 2 ) ≤ K (u 0 ) for any t ≥ 1 and w(x, z, ·; u 0 ) C 1 ([1,∞)) ≤ K (u 0 ) for any (x, z) ∈ R 2 .
Repeating the above argument, we easily get which implies that This completes the proof.
Similar to Ninomiya and Taniguchi [43,Lemma 4.3], we have the following lemma.

Lemma 3.2 There exists a positive constant
The following two lemmas establish some super-and subsolutions of (3.1).

Lemma 3.3 Letv be a supersolution to
Let v be a subsolution to (1.9) with Proof From the definition of w + and w -, we have ). For convenience, let v be either v or v. By the assumptions, for Here M is defined in (3.6) and l 2 is as in (2.6). For v < δ 1 or v > 1δ 1 , we have if we set 0 < β < ω and ρ > l 2 β . Take β > 0 and ρ > 0 such that 0 < β < ω and ρ > β+M ββ 3 + l 2 β . Then we obtain w + t + L[w + ] ≥ 0 and wt + L[w -] ≤ 0. Thus, we have proved that w + and ware a supersolution and a subsolution, respectively. This completes the proof.
To prove the asymptotical stability of the traveling curved front v * , we also need the following important auxiliary lemmas. Proof Define Then we have Here h(x, z, t) = 0 uniformly for t ≥ 0.
Since v + (x, z; ε, α) is a supersolution of (1.9), we have that w(x, z, t; v + ) is monotone decreasing in t and the limit function Lemma 3. 5 Let v * and v * be as in (2.26) and (3.17).
The proof of the lemma is similar to that of Ninomiya and Taniguchi [43,Lemma 4.6], so we omit it. The following theorem shows that the traveling curved front v * is asymptotically stable for the initial data u 0 ∈ BUC 1 (R 2 ) with u 0 ≥ v -.
Remark 3.7 Combining Theorems 2.3 and 3.6, we can complete the proof of Theorem 1.1. Theorem 3.6 also asserts that v * is a unique traveling curved front satisfying (1.10) and (1.12).

Discussion
In this paper, under assumptions (F) and (G), we establish the existence and stability of traveling curved front v * of (1.1) in R 2 for every direction θ ∈ (0, π/2) satisfying (C). For such a reaction-convection-diffusion equation, as mentioned in the first section, the planar traveling wave profile U θ of (1.1) and the corresponding wave speed c θ depend on the propagation direction θ ∈ [0, 2π). Clearly, in this paper we only consider a simple convection term (g(u)) y = ∇ · (0, g(u)), namely, it is supposed that the nonlinear convection only occurs in the y-direction. Let U θ (x cos θ + y sin θ + c θ t) be the traveling wave front of (1.1) along the direction θ ∈ (0, π/2) (or (cos θ , sin θ )). Due to such an assumption, we always have that U θ (-x cos θ + y sin θ + c θ t) is a planar traveling wave front of (1.1) along the direction πθ (or (-cos θ , sin θ )). Hence, we can prove the main results of this paper by using the method similar to those in Ninomiya and Taniguchi [43] and Wang [54]. Beyond all doubt, it is more reasonable to consider the following convection term: ∇ · h(u), g (u) .
But in this case, the function U θ (-x cos θ + y sin θ + c θ t) is no longer a traveling wave front of the equation along the direction πθ (or (-cos θ , sin θ )). Thus, the supersolution constructed in Lemma 2.2 does not work in this case and we cannot get the existence and stability of traveling curved fronts by the arguments of this paper. Therefore, to consider traveling curved fronts of (1.1) with a convection term ∇ · (h(u), g(u)) is a very interesting and difficult problem, and we leave it as a future work.
Here we also would like to give more comments on conditions (F)(iii) and (C). In fact, for every θ ∈ [0, 2π), the existence of traveling wave front U θ (x cos θ + y sin θ + c θ t) of (1.1) follows from conditions (F)(i), (F)(ii), (F)(iv), and (G). Consequently, we can get c 0 > 0 by condition (F)(iii). As discussed in Sect. 1, it follows from c 0 > 0 that there exists a subset of (0, π/2) in which every θ satisfies condition (C) (at least, there exists θ * ∈ (0, π/2) such that each θ ∈ [0, θ * ) satisfies condition (C)). On this basis, for each θ ∈ (0, π/2) which satisfies (C), we can establish the corresponding traveling curved front v * (x, y + s θ t) with speed s θ = c θ sin θ , see Theorem 1.1. Clearly, to establish the existence of traveling curved fronts by the method of this paper, the supersolution constructed in Lemma 2.2 plays a crucial role. Observing the proof of Lemma 2.2, we find that inequality (2.12) seems indispensable. Thus, condition (C) is necessary for using the method of this paper to establish the existence of traveling curved fronts. By a direct calculation, we have Under assumption (F)(iii), the inequality c θ + sup r∈[0,1] g (r) sin θ < 0 cannot hold, because the inequality implies that 1 0 f (r) dr < 0. Thus, under conditions (F) and (G), for θ ∈ (0, π/2) which does not satisfy (C), do traveling curved fronts of (1.1) exist or not? How to establish the traveling curved front of (1.1) in this case? These are very interesting questions.