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.


Introduction
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][2][3][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][6][7][8][9].For the delayed reaction-diffusion equations, Schaaf [10] and the authors [11] studied the stability of the traveling waves.Then, by applying the upper and lower solutions method combined with the squeezing technique, Wang et al. [12] 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  2 norm.Later, Mei and coauthors [15,16] obtained the globally exponential stability of the traveling waves by using the similar methods in [17] with some improvement.Recently, Mei et al. [18] gave the globally exponential stability of all noncritical wavefronts and the globally algebraic stability of critical wavefronts for a nonlocal time-delayed reactiondiffusion equations by using the weighted energy method and the Fourier transform.Immediately, Mei and Wang [19] 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 nonlinearnonlocal functional response with the initial data where  ∈ R,  ≥ 0,  > 0 is a diffusion coefficient, and the convolution is defined by Some well-known models can be obtained by choosing proper cases of ,  and K.For example, some of them can be generalized as below.
(ii) Set  = 0 and K(, ) = ( − )(); then (1) turns to the following nonlocal reaction-diffusion equation with discrete time delay In [20], 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 [21] 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 [22] 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 [24] 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 [24] 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: ( 1 ) K(, ) is a continuous nonnegative function with K(, −) = K(, ) for  ≥ 0,  ∈ R and d d < ∞ for any  ≥ 0 and  ≥ 0.
From ( 3 ), 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 R 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.

Preliminaries and Main Results
In this paper, we suppose that  > 0 represents a generic constant and   > 0 denotes a particular constant.Set  as an interval; typically  = R.Take with the norm defined by Moreover, set  > 0 as a constant and B as a Banach space; we denote by ([0, ]; B) the space of the B-valued continuous functions on [0, ].The corresponding spaces of the B-valued functions on [0, ∞) are defined similarly.Now we show the following existence result of traveling wave solutions of (1).
Then we give the following globally exponential stability of the traveling wavefronts of (1).
Theorem 2 (stability of traveling waves).Suppose that ( 1 )-( 6 ) 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  =  1 /3 and  1 is determined by Remark 3. The authors in [24] investigated the exponential stability of traveling waves of (1) when where In addition, [24] needs the following assumption: But condition ( 5 ) is not required in [24].In Section 4, we can demonstrate the advantage of our result by the same example.

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.
This completes the proof.

Applications
In this section, we will apply the content of Theorem 2 to some realistic examples.

4.1.
A Vector-Disease Model.In [3,25], the authors considered the following equation: with a positive number  =  1 /3, where  1 is determined by Remark 15. [24], under the additional condition  > (1 + (1/2)), the exponential stability of traveling waves of (96) is obtained for where Remark 16.Corollary 14 shows that if  >  > 0, the density of infectious individual tends to an endemic level  + = 1 − / when time  goes on.Further, when  ↗ , the density of infectious individual tends to 0. That is, when the recovery rate of the infected individual is close to the host-vector contact rate, this disease will eventually die out.This shows that an effective way to control the epidemic is to reduce the host-vector contact rate.Therefore, 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.

A Population Dynamic Model.
In [17], the authors considered a single species in the spatial transport field and obtained the following general reaction advection diffusion equation with the nonlocal delayed effect if the mature mortality and diffusion rates are independent of age, where Here (, ) is the density of the population of the species at time  and location ,   is the diffusion rate and   is the mortality,  is the transport velocity of the field,  ≥ 0 is the maturation time for the species,  represents the impact of premature mortality, and  demonstrates the effect of immature population dispersal rate on mature population growth rate.Liang and Wu [17] obtained the existence of the traveling wavefronts by using the upper and lower solutions when () took three different birth functions which have been widely studied in the well-known Nicholson's blowflies equation.Finally, they gave the numerical stability of travelling waves by choosing some special data.Later, Wu et al. [24] showed the exponential stability of traveling waves with faster wave speed for (105).Now we will establish the exponential stability of all noncritical traveling waves for (105).First, we need to give the following hypotheses: ( 1 ) Let  > 0, (0) = 0, () =   , () ∈  2 [0, ],   () ≥ 0,   () ≤ 0 for  ∈ [0,], () >    for  ∈ (0, ).
Clearly,  − = 0 and  + =  are two constant equilibria of (105).And the characteristic equation of ( 105  ( Remark 18.The model can be used to study the behaviour of the mature population.For example, if we take birth function () =  −  , where , ,  are positive constants and let 1 < /  <  1/ , it is easy to verify that, for  >  * +, all noncritical traveling waves connecting 0 and ((1/) ln(/  )) 1/ are exponential stable.That is, the density of the population tends to a constant ((1/) ln(/  )) 1/ as time becomes large.The results show that the density of the population will eventually reach a stable state.