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Entire solutions for a reaction–diffusion equation with doubly degenerate nonlinearity
Advances in Difference Equations volume 2018, Article number: 147 (2018)
Abstract
This paper is concerned with the existence of entire solutions for a reaction–diffusion equation with doubly degenerate nonlinearity. Here the entire solutions are the classical solutions that exist for all \((x,t)\in \mathbb{R}^{2}\). With the aid of the comparison theorem and the supsub solutions method, we construct some entire solutions that behave as two opposite traveling front solutions with critical speeds moving towards each other from both sides of xaxis and then annihilating. In addition, we apply the existence theorem to a specially doubly degenerate case.
Introduction
In this paper, we consider the following scalar reaction–diffusion equation:
where f satisfies \((A)\) \(f\in C^{2} ([0,2] ), f(0)=f(1)=0, f'(0)=f'(1)=0\), \(f'(s)>0, f'(1s)<0\) for small \(s>0\), and \(f(u)>0\) for \(u\in(0,1)\).
From \((A)\), it is easy to see that \(u=0, u=1\) are two constant equilibria of (1.1).
In practical applications, traveling wave solutions can well explain oscillations and finite velocity propagation phenomena in nature. Thus, in recent years, the existence and the stability of traveling wave solutions of (1.1) have been extensively studied by many scholars, see [1–5] and the references therein. A traveling wave solution of (1.1) is a solution of the form \(u(x,t)=\phi(x+ct)\) satisfying
where \(':=\frac{{\mathrm {d}}}{{\mathrm {d}}z}, z=x+ct\) and c is the wave speed. Moreover, a monotone and bounded traveling wave solution is called a traveling front solution. In [3], Hou et al. investigated the existence, uniqueness, asymptotic behavior as well as the stability of the traveling front solutions for (1.1).
However, it is not enough to understand the dynamical structure of solutions of reaction–diffusion equations by only considering traveling front solutions. From the dynamical view, the study of entire solutions is essential for a full understanding of the transient dynamics and the structures of the global attractor as mentioned in [6]. For example, the ωlimit sets of bounded solutions of (1.1) and global attractor are comprised of entire solutions as mentioned in [6, 7]. In recent years, the existence of entire solutions is widely discussed. Firstly, when f satisfies
then (1.1) becomes the famous Fisher–KPP equation (monostable case). For this equation, Hamel and Nadirashvili in [8] proved the existence of entire solutions by the comparison theorem and supersub estimates, which consists of traveling front solutions and solutions to the diffusionfree equations. Moreover, they also pointed out that the entire solutions to (1.1) depend only on t and traveling wave solutions are typical examples of entire solutions, and they showed various entire solutions of (1.1) with (1.3) in their subsequent paper [9]. While for both \(f'(0)<0\) and \(f'(1)<0\) (bistable case), Yagisita in [10] revealed that the annihilation process is approximated by a backward global solution of (1.1), which is the entire solution. We call this kind of entire solutions annihilating entire solutions, and in the following, without special indication, we use entire solutions to represent annihilating entire solutions for short. For Allen–Cahn equation
with \(a\in(0,1)\), as a special example in [10], Fukao et al. in [11] gave a proof for the existence of entire solutions by using the explicit expression of the traveling front and the comparison theorem. Later, Guo and Morita in [12] extended the results in [8] and [10] to more general cases including a discrete KPP equation by the supersub solutions method and the comparison theorem. At the same time, Chen and Guo in [13] proved the existence of entire solutions by showing a different technique which used only one function to construct a pair of deterministic supersub solutions. Moreover, Chen et al. in [14] also used the method in [13] to construct entire solutions based on standing waves. Furthermore, in [6], Morita and Ninomiya constructed two kinds of merging entire solutions. Very recently, Wang in [15] investigated the entire solutions for the degenerate Fisher equation by considering two traveling front solutions with critical speeds. And Zhang et al. in [16] dealt with the frontlike entire solution of a classical nonlocal dispersal equation with ignition nonlinearity, where the dispersal kernel function may not be symmetric.
However, little is known about the entire solutions of reaction–diffusion equations with doubly degenerate nonlinearities. Thus, encouraged by [6, 12, 13, 15, 16], in this paper we will investigate the entire solutions of (1.1) under assumption \((A)\). Moreover, from [17] and the references therein, we know that the doubly degenerate reaction–diffusion equations usually mean that the diffusion term is assumed to be doubly degenerate. Here, from assumption \((A)\), we mainly focus on the reaction–diffusion equation with doubly degenerate nonlinearity.
The rest of this paper is arranged as follows. In Sect. 2, we show the existence and the asymptotic behaviors of traveling front solutions of (1.1) and give some preparations that will be used to construct the supsub solutions. An entire solution to (1.1) can be obtained by considering two traveling front solutions with critical speeds that come from both sides of the xaxis in Sect. 3. Finally, we apply our results to an example.
Preliminaries
In this section, we will give some relevant preparations in order to obtain the main conclusions later. First of all, we state the definitions of supersolution and subsolution of (1.1) as follows.
Definition 2.1
Set \(\Omega:=\mathbb{R}\times[r, R]\) for some \(R>r\). \(\underline {u}(x,t)\) is called a subsolution of (1.1) in Ω if \(\underline{u}(x,t)\leq u(x,t)\ ((x,t)\in\Omega)\) for every solution of (1.1) defined in Ω such that \(\underline{u}(x,r)\leq u(x,r)\) \((x\in\mathbb{R})\). If \(\underline{u}(x,t)\) is a subsolution of (1.1) in \(\mathbb {R}\times[r,T]\) for any \(r<T\), then \(\underline{u}(x,t)\) is called a subsolution of (1.1) in \(\mathbb{R}\times[\infty,T]\) for some \(T\geq0\). Similarly, we can define a supersolution by reversing the inequalities.
Let \(\mathcal{F}(u)=u_{t}u_{xx}f(u)\). We notice that a bounded function \(u(x,t)\) is a subsolution of (1.1) in \(\mathbb{R}\times(\infty , T)\ (T\geq0)\) if \(\mathcal{F}(u)\leq0 \mbox{ for } (x,t)\in\mathbb{R}\times(\infty, T)\), while it is a supersolution if \(\mathcal{F}(u)\geq0\).
From [3], we can obtain the following result, which shows the existence and the asymptotic behavior of traveling front solutions of (1.1).
Lemma 2.2
([3])
Suppose that \((A)\) holds, then there is a unique positive number \(c^{*}\) such that, for any \(c\geq c^{*}\), (1.2) has a monotonically increasing solution \(\phi(z)\), while for \(c^{*}>c>0\), there is no positive solution for (1.2). Moreover, the solutions of (1.2) have the following asymptotic behaviors:
where \(B>0\) is a constant, \(H_{0}=F_{0}^{1}, F_{0}=\int_{u_{0}}^{u}\frac{{\mathrm {d}}s}{f(s)},0< u< u_{0}\) for some small \(u_{0}\), \(H_{1}=F_{1}^{1},F_{1}=\int _{u_{1}}^{u}\frac{{\mathrm {d}}s}{f(s)},u_{1}< u<1\) and \(u_{1}\) is close to 1.
By Lemma 2.2, it is obvious that there are positive constants \(r_{i}\) (\(i=1,2\)), α and η such that
where \(\alpha\leq c^{*}\) and \(\phi(z)\) is the traveling front solution with critical speed. For simplicity of statements, we denote c as \(c^{*}\) in the following.
In order to establish the supersolution of (1.1), we should consider the following ordinary differential problem:
where
and \(L:=\max_{u\in[0,2]} \vert f''(u) \vert \). From direct calculation, we can obtain the solution of (2.4) explicitly as
and
Moreover, there exists \(K>0\) so that for \(t\leq0\)
The existence of entire solutions
In this section, we discuss the existence of entire solutions of (1.1). Firstly, we construct the supersolution.
Lemma 3.1
Assume that \((A)\) holds, and let \(p(t)\) be the solution of (2.4). Then the function
is a supersolution of (1.1) for \(t\leq0\), where \(\tilde{\phi}(x+p(t))=\phi(x+p(t))\).
Proof
For simplicity, we use ϕ and ϕ̃ instead of \(\phi (x+p(t))\) and \(\tilde{\phi}(x+p(t))\). Then we have
where
Now we need to divide \(\mathbb{R}\) into three parts to estimate \(G(x,t)\).
(i) For \(p\leq x\leqp\), then \(x+p\leq0, x+p\leq0\). Noting that \(f(0)=0,f\in C^{2}([0,2])\) and \(\phi,\tilde{\phi}\in[0,1]\), we can get
Combining with (2.2) and (2.3), it follows that
(ii) For \(x\leq p\), then \(x+p\leq0, x+p\geq0\). Now we can modify \(f'(u)\) for \(u\in(1,2]\) so that \(f'(u)< f'(0)=0\) for \(u\in(1,2)\). Noting hypothesis \((A)\), it follows that there exists \(\delta\in(0,1)\) such that
Factually, (3.3) can be assured by translating \(\phi(z)\) along zaxis. In the following, we may assume that
Then it follows from (3.3) and (3.4) that
Then
(iii) For \(x\geqp\), then \(x+p\geq0, x+p\leq0\). Noting that \(G(x,t)=G(x,t)\) and (3.5), we get
Then, combining (2.5), (3.1), (3.2), (3.5), and (3.6), we have
The proof is complete. □
Theorem 3.1
Assume that \((A)\) holds. Let ϕ and ϕ̃ be traveling front solutions of (1.1) with the critical speed c. Then, for arbitrarily given constants \(\theta_{1}, \theta_{2}\), there exists an entire solution \(u(x,t)\) which satisfies
Moreover, if \(c\alpha>N=\max_{u\in[0,2]} \vert f'(u) \vert \), then this entire solution is unique. Furthermore, this solution satisfies

(i)
\(\frac{\partial u(x,t)}{\partial t}>0\) for all \((x,t)\in\mathbb{R}\times\mathbb{R}\);

(ii)
\(\lim_{t\rightarrow\infty}\sup_{x\in [\alpha, \beta]} \vert u(x,t) \vert =0\) with \(\alpha, \beta\in\mathbb{R}\) and \(\alpha<\beta\);

(iii)
\(\lim_{ \vert x \vert \rightarrow\infty}\sup_{t\in[S, \infty )} \vert u(x,t)1 \vert =0 \) with \(S\in\mathbb{R}\).
Proof
Define
It is obvious that
Combining with the comparison principle, it yields that there exists a solution \(\tilde{u}(x,t)\) of (1.1) which satisfies
Consider the following problem:
It is easy to see that there is a unique solution \(u(x,t)\) of (3.9) which also satisfies \(\underline{u}(x,t)\leq u(x,t)\leq1\) for any \((x,t)\in\mathbb{R}\times[0,+\infty)\). Let
Then we can obtain an entire solution of (1.1) which satisfies
and
Next we will prove (3.7). Firstly, we prove
Then, for \(x\geq0\) and \(t\leq0\), we can get
For \(x\leq0\) and \(t\leq0\), similarly, we obtain
For any given \(\theta_{1}, \theta_{2}\), by choosing
we can obtain the entire solution of (1.1) as \(\hat {u}(x,t)=u(x+\mu, t+\tau)\).
Now we discuss the uniqueness. Set \(u_{1}(x,t), u_{2}(x,t)\) be two entire solutions of (1.1). Suppose that \(u_{1}(x,t)\geq u_{2}(x,t)\). Noting that \(\max\{u_{1}(x,t), u_{2}(x,t)\}\) is a subsolution, there exists an entire solution \(\hat{u}(x,t)\) satisfying
For any given \((x,t),t\leq0\), let \(\tau\leq t\). Setting \(v(x,t):=u_{1}(x,t)u_{2}(x,t)\), we have
Then
Furthermore, from the above estimates it also follows that there exists \(D>0\) such that
Thus we can get
Since τ can be made arbitrarily small and \(c\alpha>N\), we can obtain that \(v(x,t)=0\) for \(x\in\mathbb{R}\). This completes the proof. □
Next we give another type of entire solutions. Suppose that \(\zeta(t)\) is a solution of the following ordinary differential equation:
where N is defined in Theorem 3.1 and \(0<\zeta(t)<1\).
Theorem 3.2
Assume that (A) holds. Define \(\rho(t):=\rho_{0}e^{Nt}\), and let \(\rho_{0}>0\) satisfy
where \(R_{0}\) is a positive number. For given \(c_{1}, c_{2}\geq c^{*},\nu_{1}, \nu_{2}\in\mathbb{R}\) and positive constants \(k_{j}\ (j=1,2),R_{1}\) and T, there exist monotone increasing functions \(p_{j}(t)\ (j=1,2)\) satisfying
such that \(\bar{u}_{ik}(x,t):=\chi_{i}\phi(x+p(t))+\chi_{k}\tilde{\phi }(x+p(t))+\rho(t)\ ((i,k)=(1,0),(0,1),(1,1))\) are supersolutions for \(t\in(\infty, T]\), where \(\chi_{i}=i\ (i=0,1)\). Furthermore, for \((x,t)\in\mathbb{R}\times (\infty,T]\), there are entire solutions \(u_{ik}(x,t),\ (i,k)=(1,0),(0,1),(1,1)\) of (1.1) which satisfy
Proof
We just need to prove that \(\bar{u}_{ik}(x,t)\) is a supersolution of (1.1). Now set
Similarly, we can have that
where
If \(x\leq0\), we can get
Noting that \(\rho'=N\rho\) and
then it follows that
Similarly, we can prove that \(\mathcal{F}(\bar{u})\geq0\) if \(x\geq0\). That is, \(\bar{u}(x,t)\) is a supersolution of (1.1). The rest of the proof is similar to that of [8]. □
Applications
For the double degenerated generalized Fishertype equation,
where p and q are not necessarily integers. Now we show the conclusion of the existence of traveling wave front solutions of (4.1).
Lemma 4.1
([4])
If \(p>1,q>1\), there exists \(c^{*}(p,q)>0\) such that, for any \(c\geq c^{*}(p,q)\), there are traveling front solutions \(\phi_{c}(z)\ (z=xct)\) connecting \(u=0\) and \(u=1\) which satisfy
It is easy to verify that \(f(u)=u^{p}(1u)^{q}\) satisfy assumption \((A)\) in Theorem 3.1. Thus we have the following result for (4.1).
Theorem 4.1
Assume that \(p>1, q>1\). Let ϕ and ϕ̃ be traveling front solutions of (4.1) with the minimum speed c. Then, for arbitrarily given constants \(\theta_{1}, \theta_{2}\), there exists an entire solution \(u(x,t)\) which satisfies
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Acknowledgements
The authors would like to thank the referee and the editor for their valuable comments which led to improvement of this work. This work was supported by the project of 12th FiveYear planning of Education Science in Shanxi (GH15026) and the Fundamental Research Funds for the Central Universities (ZY1821).
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Correspondence to Xiaocui Li.
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MSC
 35K57
 35B40
Keywords
 Entire solutions
 Traveling wave solutions
 Doubly degenerate nonlinearity