Exponential Stability of Traveling Waves for a Reaction Advection Diffusion Equation with Nonlinear-Nonlocal Functional Response
The stability of a reaction advection diffusion equation with nonlinear-nonlocal functional response is concerned. By using the technical weighted energy method and the comparison principle, the exponential stability of all noncritical traveling waves of the equation can be obtained. Moreover, we get the rates of convergence. Our results improve the previous ones. At last, we apply the stability result to some real models, such as an epidemic model and a population dynamic model.
As an important nonlinear parabolic equation, reaction (advection) diffusion equations are usually used to describe the development of practical problems related to time and space distribution. The large number of problems involved in reaction-diffusion equations arises from a large number of mathematical models in chemistry, physics, ecology, biology, and infectious diseases, for instance, [1–4]. The traveling wave solutions of reaction-diffusion equations refer to the solutions of the shape, such as , where the wave velocity is a real constant. In practical applications, traveling wave solutions can well explain oscillations and finite velocity propagation phenomena in nature such as the change of the concentration of a reactant in a chemical reaction, the invasion of species in biology, the mutual transformation of two states of matter in condensed matter physics, the propagation of nerve impulses in neural networks, and the spread of infectious diseases. The stability of traveling wave solutions based on the existence of traveling wave equations is also very important and has great theoretical and practical value. In the phase transition process, a necessary condition that the state change can be observed is that the traveling wave solutions are stable. For example, in the sense of epidemiology, the traveling wave solutions describe the transition from a disease-free equilibrium to an endemic equilibrium, while the existence and nonexistence of nontrivial traveling wave solutions suggest whether or not the disease can spread. The results will help predict the developing trend of infectious diseases, identify the key factors for the spread of infectious diseases, and seek the best strategies to prevent and control the spread of infectious diseases.
Hence, the study of traveling wave solutions has become a very interesting problem, in which the stability of traveling waves is significant and difficult, and this topic attracts enormous attention. For the stability of reaction-diffusion equations without time delay, there is a great deal of literature, for example, [5–9]. For the delayed reaction-diffusion equations, Schaaf  and the authors  studied the stability of the traveling waves. Then, by applying the upper and lower solutions method combined with the squeezing technique, Wang et al.  presented the exponential stability of traveling wavefronts for a reaction advection diffusion equation with spatiotemporal delays under the bistable assumptions. Mei et al. [13, 14] proved that the large waves were exponentially stable when an initial solution is close enough to the traveling wave solution under a weighted norm. Later, Mei and coauthors [15, 16] obtained the globally exponential stability of the traveling waves by using the similar methods in  with some improvement. Recently, Mei et al.  gave the globally exponential stability of all noncritical wavefronts and the globally algebraic stability of critical wavefronts for a nonlocal time-delayed reaction-diffusion equations by using the weighted energy method and the Fourier transform. Immediately, Mei and Wang  obtained the similar results for a class of nonlocal time-delayed Fisher-KPP type reaction-diffusion equations in -dimensional space by employing the similar methods, in which the convergence rates were more accurate than previous results.
Motivated by these previous works, in this paper, we consider the stability of the traveling waves of the following reaction advection diffusion equation with nonlinear-nonlocal functional response with the initial data where , is a diffusion coefficient, and the convolution is defined by Some well-known models can be obtained by choosing proper cases of and . For example, some of them can be generalized as below.
(i) Set and ; then (1) turns to the following local reaction-diffusion equation with discrete delay Schaaf  studied the stability of the traveling waves of (4) by a phase plane analysis method. Then, the authors in  showed the global asymptotic stability, Lyapunov stability, and uniqueness of traveling waves of (4) by using the elementary super- and subsolution comparison and squeezing methods.
(ii) Set and ; then (1) turns to the following nonlocal reaction-diffusion equation with discrete time delay In , the existence of traveling wavefronts of (5) was investigated by using the sub- and supersolution method. The authors also proved the asymptotic stability and uniqueness of traveling wave fronts by applying the comparison principle and squeezing technique. Later, Wang  proved the existence of the traveling wave solutions of (5) by using the upper and lower solutions method and the Schauder fixed point theorem when the term may not be monotone or quasimonotone. Recently, the authors  got the exponential stability of all noncritical traveling waves of (5) and the algebraic stability of critical traveling waves of (5) by using the weighted energy method coupled with the comparison principle.
It follows from the above examples that these local or nonlocal equations with discrete time delay are studied intensively. In some cases, a discrete delay is a good approximation, but the distributed delay is necessary in others. On the other hand, reaction advection diffusion equations have been usually applied to describe the dispersion courses in the mobile media such as fluids; see, for example, [12, 23]. In fact, Wu  proved the exponential stability of traveling wavefronts with large wave velocity of (1) by using the comparison principle and the technical weighted energy method. By choosing a different weighted energy function from  and adopting different estimates, then we can get the stability of all noncritical traveling wavefronts of (1), not just the traveling waves with large wave velocity.
Now we impose some assumptions on (1) as follows:() is a continuous nonnegative function with for and , for any and .() For any , where satisfies .() Let , , , for for any , where and .() for any .() .() .
From , it is easy to see that (1) has two constant equilibria , where Throughout this paper, a traveling wave solution of (1) always refers to a pair , where is a function on satisfying the following ordinary differential equation: We call the traveling wave speed.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries and present our main result on the exponential stability of traveling waves of (1). In Section 3, we will give the proof the main result by using the weighted energy method. In Section 4, we apply our results to some models, such as an epidemic model and a population dynamic model.
2. Preliminaries and Main Results
In this paper, we suppose that represents a generic constant and denotes a particular constant. Set as an interval; typically . Take with the norm defined by Moreover, set as a constant and as a Banach space; we denote by the space of the valued continuous functions on . The corresponding spaces of the valued functions on are defined similarly.
Now we show the following existence result of traveling wave solutions of (1).
Theorem 1 (, existence of traveling waves). Assume that hold; there exist numbers and satisfying where For every , (1) has a monotone traveling wave solution such that and .
From the assumptions we can introduce that . Then by a simple calculation similar to that of  we can get the following conclusions. When , has two positive real roots and with which also satisfies When , it holds that
Then, for , we define a weight function related to such a number , with a sufficiently large number , where . Obviously for all and as .
Then we give the following globally exponential stability of the traveling wavefronts of (1).
Theorem 2 (stability of traveling waves). Suppose that hold. For the traveling wave of (1) and with speed , if the initial data holds and the initial perturbation is ; then the solution of (1) and (2) satisfies If , the solution converges to the traveling wave exponentially with a positive number , where and is determined by
Remark 3. The authors in  investigated the exponential stability of traveling waves of (1) when where In addition,  needs the following assumption: But condition () is not required in . In Section 4, we can demonstrate the advantage of our result by the same example.
3. Proof of the Stability
For the stability result, we need the following boundedness and the comparison principle for (1). The proofs are similar to those of [16, 24], so we omit them here.
Lemma 4 (boundedness). Assume hold and the initial data satisfy thus the solution of the Cauchy problem (1) and (2) exists uniquely and satisfies
Lemma 5 (comparison principle). Set and as the solutions of (1) and (2) with the initial data and , respectively. When thus
If satisfies letting then Set and as the corresponding solutions of (1) and (2) with initial data and determined in (27); that is, Thus by Lemma 5 we get
Now we need the following three steps to get the stability result.
Step 1. The convergence of to is as follows.
For every , take and From (28) and (30) we get Then it can be checked that defined in (31) satisfies with the initial data where
Lemma 6. For , by choosing such that where then, for , the following holds:
Proof. Assume the solution of (33)-(34) is mollifying enough. Multiplying (33) by , where is defined in (37), we get Integrating both sides of (40) over with respect to and , thus For , applying Taylor’s formula to (36) and noting ; then where is some function between and and is some function between and .
For the third term on the right-hand side of (41) by Fubini’s theorem and making the change of variables , it follows that Substituting (42) and (43) into (41), it follows that where Since and noting , the following holds: Using the fact that, for , then we can choose a small such that For the second term on the right-hand side of (44), we have Substituting (48)-(49) into (44) we can get that is, This completes the proof.
Lemma 7. For every , there holds that
Proof. In view of for , it can follow from (50) that, for any , and letting , we obtain Since , for , we have Multiplying (33) by and integrating it over with respect to and , it yields that By employing the Cauchy inequality for , we get For the second term on the right-hand side of (57), by making the change of variables and using (55), we have For the first term on the right-hand side of (57), by using (55), we have Substituting (58), (59) into (57), we obtain Noticing that and using (60) to (56), it yields that Because of and , we can select as follows: Then we have Choose large enough such that if , we can obtain Because of , for , we have Thus substituting (65) into (61) and setting we can obtain This completes the proof.
Next, we differentiate (33) with respect to , multiply it by , and integrate it over with respect to and . Combining with Lemma 7, we can get the following energy estimate. Here we omit the details of the proof.
Lemma 8. If and , the following holds:
Lemma 9. If and , the following holds:
Proof. Take . Thus By the Hölder inequality, we can get It follows from (69) and (71) that that is, From (67) we have Noting that if and combining with (39), we can get Then it is obvious from (73)–(75) that