Traveling waves for combustion reaction-diffusion-convection equations:
the full range of wave speeds
May 30, 2026
We consider traveling wave solutions to a reaction–diffusion–convection equation with a combustion-type reaction term. While a necessary condition for the existence of traveling waves is \(c \geq H^*:=\sup_{0<u \leq \theta} \big(-\frac{1}{u} \int_0^u h(\sigma)\,d\sigma \big)\), where \(c\) denotes the wave speed, \(\theta\in(0,1)\) the ignition threshold, and \(h\) the convective term, the available results in [1], [2] establish existence and nonexistence only under the restriction \(c \geq -\min_{u\in[0,1]} h(u)\). In this note, we close this gap by covering the entire range \(c\ge H^*\).
Key words: traveling waves, combustion reaction, reaction-diffusion-convection equations
We study traveling wave solutions to the reaction-diffusion-convection equation \[\label{rdc95com} v_t=\left[d(v)|v_x|^{p-2}v_x\right]_x+h(v)v_x+g(v), \quad (x,t) \in \mathbb{R} \times [0,+\infty).\tag{1}\] Here \(p>1\), \(h \in C[0,1]\) is a convective velocity, \(d \in C^1(0,1)\) is a diffusion coefficient which may degenerate or become singular at one or both endpoints, and \(g \in C[0,1]\) is a combustion-type reaction term, i.e., \[\label{r} \text{g(v)=0 in [0,\theta]}, \quad \text{g(v)>0 in (\theta, 1)}, \quad g(1)=0\tag{2}\] for some \(\theta \in (0,1)\).
Reaction–diffusion–convection equations with combustion-type nonlinearities arise in models of flame propagation and interface dynamics, and traveling wave solutions play a fundamental role in their analysis. See [1], [2] and the references therein for a detailed discussion of the model.
When seeking a traveling wave solution, we formally set \(v(x,t)=u(x-ct)\), where \(u\) is the wave profile and \(c\) denotes the wave speed. Substituting this form, with \(\xi=x-ct\), into 1 we obtain the profile equation \[\label{pe} (d(u)|u'|^{p-2} u')' +(c+h(u) )u' +g(u) =0,\tag{3}\] where \(\prime=\frac{d}{d\xi}\). The objective of this note is to study monotone solutions of 3 satisfying the boundary conditions \[\label{bc} \lim_{\xi \rightarrow -\infty} u(\xi) =1 \quad and \quad \lim_{\xi \rightarrow \infty} u(\xi) = 0.\tag{4}\] Throughout the paper, we denote \[h_m:=\min_{u \in [0,1]}h(u), \quad h_M:=\max_{u \in [0,1]}h(u), \quad \quad H(u):=\int_0^u h(\sigma) d\sigma. \notag\]
The existence and nonexistence of such solutions have been recently studied in [1], extending and generalizing the results of [2], which were investigated for \(p=2\), to the present setting. However, their results in [1], [2] are obtained under the restriction \[\label{res} c \geq -h_m,\tag{5}\] where the effective speed \(c+h(u)\) remains nonnegative for all \(u \in[0,1]\), so that the convective effect never acts against the propagation of the traveling wave.
On the other hand, due to the combustion structure \(g(u)=0\) in \([0,\theta]\), one obtains the necessary condition (see 10 ) \[\label{nc} c \geq H^*, \quad H^*:=\sup_{0<u \leq \theta} \Big(-\frac{H(u)}{u} \Big).\tag{6}\] Since \(H^* \leq -h_m\), the restriction 5 imposed in [1], [2] automatically implies 6 . For this reason, the quantity \(H^*\) does not explicitly appear in their analysis, and neither existence nor nonexistence results are available for wave speeds in the range \[\label{regime} H^* \leq c < -h_m.\tag{7}\] The main purpose of the present note is to fill this gap. More precisely, we derive a sufficient condition for nonexistence for arbitrary wave speeds, as well as a sufficient condition ensuring that traveling waves may still exist even when the convective effect locally opposes the propagation of the wave.
As shown in [1], the monotonicity of solutions to 3 –4 is guaranteed. As a consequence, the problem can be reduced to a first-order boundary value problem (see 8 ), and all arguments can be carried out at this level. The restriction 5 imposed in [1], [2] provides several technical advantages in the analysis of the associated first-order Cauchy problem, including uniqueness and monotonicity properties. In contrast, in our previous work on the monostable and bistable case ([3], [4]), we showed the first-order problem can still be analyzed even when the sign of \(c+h(u)\) changes. By adapting this approach to the present combustion setting, we are able to remove the restriction 5 .
In recent years, traveling wave solutions for reaction–diffusion(–convection) equations with discontinuous density-dependent coefficients have received considerable attention (see [3]–[5] and references therein). In the present combustion setting, however, we work under the same regularity assumptions as in [1], since no substantial additional difficulties arise from lower regularity as long as the convection term \(h\) is bounded and continuous at \(0\). Accordingly, the sole focus of this paper is to fill the gap in the range of wave speeds.
In this section, we briefly recall several preliminaries from [1], [3], [4] that will be used throughout the paper, including the notion of solutions, the equivalent first-order boundary value problem, and the associated terminal value problem.
First, solutions of 3 –4 are understood in the following sense.
Definition 1 ([1]). A continuous function \(u : \mathbb{R} \rightarrow [0,1]\) is a solution of 3 –4 if
\(u \in C^1 (I_u)\), where \(I_u := \{ \xi \in \mathbb{R}: 0 <u(\xi) <1 \}\), and 3 holds at every \(\xi \in I_u\);
the function \(\xi \rightarrow d(u(\xi))|u'(\xi)|^{p-2} u'(\xi)\) is continuous on \(\mathbb{R}\) and \(d(u(\xi))|u'(\xi)|^{p-2} u'(\xi) \rightarrow 0\) as \(u(\xi) \rightarrow 0\) and \(u(\xi) \rightarrow 1\);
(boundary condition) \(u(\xi) \rightarrow 1\) as \(\xi \rightarrow -\infty\) and \(u(\xi) \rightarrow 0\) as \(\xi \rightarrow +\infty\).
It was shown in [1] that every solution is strictly decreasing on the interval where \(0<u<1\). Following [1], [3], [4], we introduce \[y(u):=w(u)^{p'}, \quad w(u):=-d(u)|u'|^{p-2}u'. \notag\] Then the problem 3 –4 reduces to the first-order boundary value problem \[\label{fode} \begin{cases} y'(u) = p' \left[ (c+h(u)) (y^+ (u))^{1/p} -f(u) \right], \quad u \in (0,1), \\ y(0)=0=y(1), \end{cases}\tag{8}\] where \(y^+(u):=\max \{y(u),0\}\), \(p'=p/(p-1)\), and \(f(u) = d(u)^{p'-1}g(u)\).
According to [1], solutions \(u\) of 3 –4 are in one-to-one correspondence with positive solutions \(y \in C^1(0,1) \cap C[0,1]\) of the first-order problem 8 . Therefore, throughout this note, we study the existence and nonexistence of traveling waves entirely through the analysis of positive solutions to 8 .
Since \(f \equiv 0\) on \([0, \theta]\), any positive solution \(y\) of 8 satisfies \[y'(u) = p' (c+h(u)) y(u)^{1/p}, \quad u \in (0,\theta), \notag\] which is separable. Using \(y(0)=0\), we obtain \[\label{fone1} y(u)^{1/p'}=cu+H(u), \quad u \in [0, \theta].\tag{9}\] Since \(y>0\) on \((0,1)\), \[\label{nc1} cu+H(u)>0 \quad \text{for all } u \in(0,\theta],\tag{10}\] from which the necessary condition 6 follows immediately. We also note that letting \(u \to0+\) in 10 yields \(c \geq -h(0)\), which is the necessary condition mentioned in [1], [2]. While those works are restricted to the range 5 , our analysis focuses on wave speeds below \(-h_m\). Consequently, the sharper condition 6 plays a crucial role in the present work.
Lastly, recalling \(f>0\) on \((\theta, 1)\), we consider the associated terminal value problem (TVP) \[\label{tvp} \begin{cases} y'(u) = p'\left[(c+h(u))(y^+(u))^{1/p}-f(u)\right], \quad u \in(0,1),\\ y(1)=0, \end{cases}\tag{11}\] and denote its solution by \(\hat{y}_c\) for each \(c \in {\mathbb{R}}\). According to [3], [4], \(\hat{y}_c\) is positive on \((\theta,1)\), unique on every interval where it remains positive, and depends continuously on \(c\). Moreover, if \(c_1<c_2\), then \[\label{mon} \hat{y}_{c_1}(u)>\hat{y}_{c_2}(u)\tag{12}\] whenever both solutions remain positive. In addition, under the assumption \[\label{mu95com} \mu:=\sup_{u \in (\theta, 1)} \frac{f(u)}{(u-\theta)^{p'-1}} <\infty,\tag{13}\] it was shown in [3], [4] that there exists a threshold wave speed \[\label{cBcom} c_B\in\big[-h(\theta),\, -h_m+(p')^{1/p'}p^{1/p}\mu^{1/p'}\big]\tag{14}\] such that the problem 11 on \([\theta,1]\), together with the condition \(y(\theta)=0\), admits a unique positive solution if and only if \[\label{cB} c \geq c_B.\tag{15}\] The existence of this threshold will play a crucial role in proving the nonexistence of positive solutions to 8 . Indeed, if \(c \geq c_B\), then \(\hat{y}_c(\theta)=0\), and hence no positive solution of 8 can exist on \((0,1)\).
We begin with the nonexistence of traveling wave solutions to 3 –4 under the necessary condition 6 , which in turn implies nonexistence for all wave speeds.
Theorem 1 (Nonexistence). Let 13 hold and assume that \[\label{nonex3} H(\theta) \geq -H^*+(1-\theta)(-h_m+(p')^{1/p'}p^{1/p}\mu^{1/p'}).\tag{16}\] Then the problem 8 has no positive solution for any \(c\geq H^*\); in particular, it admits no positive solution for any \(c \in {\mathbb{R}}\).
Proof. We argue by contradiction. Suppose that 8 admits a positive solution \(y_c\) for some \(c\geq H^*\). Since \(y_c\) also solves the TVP 11 on \([\theta,1]\), the threshold property 15 implies that necessarily \(c<c_B\). Divide 8 by \(y_c(t)^{1/p}\) and integrate over \((\theta,1)\) to obtain \[\label{non1} y_c(\theta)^{1/p'} =-c(1-\theta)-H(1)+H(\theta) +\int_{\theta}^1 \frac{f(u)}{y_c(u)^{1/p}}\,du.\tag{17}\] Applying the same identity to \(\hat{y}_{c_B}\) and using \(\hat{y}_{c_B}(\theta)=0\), we have \[\label{non2} \int_{\theta}^1 \frac{f(u)}{\hat{y}_{c_B}(u)^{1/p}}\,du = c_B(1-\theta)+H(1)-H(\theta).\tag{18}\] Since \(c<c_B\) and \(f>0\) on \((\theta, 1)\), the monotonicity 12 yields \[\int_{\theta}^1 \frac{f(u)}{y_c(u)^{1/p}}\,du = \int_{\theta}^1 \frac{f(u)}{\hat{y}_c(u)^{1/p}}\,du < \int_{\theta}^1 \frac{f(u)}{\hat{y}_{c_B}(u)^{1/p}}\,du. \notag\] Using 14 , \(c \geq H^*\), and the assumption 16 , it follows from 17 –18 that \[\label{upper95y} y_c(\theta)^{1/p'} < (1-\theta)(c_B-c) \leq (1-\theta)\big(-h_m+(p')^{1/p'}p^{1/p}\mu^{1/p'}-H^*\big) \leq H^*\theta +H(\theta).\tag{19}\]
On the other hand, again using \(c \geq H^*\), we deduce from 9 that \[y_c(\theta)^{1/p'} \geq H^*\theta+H(\theta), \notag\] which contradicts 19 . This completes the proof. ◻
Remark 1. In [1], nonexistence is proved for \(c \geq -h_m\) under the assumptions \[\label{assDZnon} \int_0^1 f(u) \, du < \infty \quad \text{and} \quad H(\theta) > \theta h_m + \Big(p'\int_0^1 f(u) \, du \Big)^{1/p'}.\tag{20}\] Since our result yields nonexistence for any real value of \(c\), the assumptions 13 and 16 imply 20 . Indeed, by 13 , \[\int_0^1 f(u)\,du = \int_{\theta}^1 \frac{f(u)}{(u-\theta)^{p'-1}}(u-\theta)^{p'-1}\,du \leq \mu \int_{\theta}^1 (u-\theta)^{p'-1}\,du = \frac{\mu}{p'}(1-\theta)^{p'} < \infty. \notag\] Since \(p>1\) and \(H^* \leq -h_m\), it follows from 16 that \[\theta h_m+\Big(p'\int_0^1 f(u)\,du\Big)^{1/p'} < \theta h_m+(1-\theta)(p')^{1/p'}p^{1/p}\mu^{1/p'}-H^*-h_m \le H(\theta). \notag\]
Example 1. Theorem 1 is particularly relevant when \(H^*<-h_m\), since otherwise the nonexistence result in [1] already applies. For example, let \(p=p'=2\), \(\theta=\frac{1}{2}\), \(d(u) =1\) on \([0,1]\), \[g(u)= \begin{cases} 0, & 0\le u \le \frac{1}{2},\\ \frac{1}{16}\left(u-\frac{1}{2}\right)(1-u), & \frac{1}{2}<u \le 1, \end{cases} \qquad \text{and} \qquad h(u)= \begin{cases} 8u, & 0\le u\le \frac{1}{2},\\ 9-10u, & \frac{1}{2}<u \le 1. \end{cases} \notag\] Then \(H^*=0 < 1=-h_m\), and 16 holds. Thus no traveling waves exist for any wave speed.
Recall that the solution \(\hat{y}_c\) of 11 is positive on \((\theta,1)\). For each \(c \geq H^*\), let \((u_c,1)\subset(0,1)\) denote the maximal interval such that \[\hat{y}_c(u)>0 \quad \text{on (u_c,1)}. \notag\] In what follows, we regard \(\hat{y}_c\) as a positive solution of 11 restricted to the interval \((u_c,1)\): \[\label{tvp95ex} \begin{cases} y'(u) = p'\left[(c+h(u))y(u)^{1/p}-f(u)\right], \quad u \in(u_c,1),\\ y(1)=0. \end{cases}\tag{21}\]
We now state the existence of positive solutions to 8 , recalling \(\kappa(p)\) introduced in [1]: \[\begin{align} \label{kappa} \kappa(p) &= \left\{ \begin{array}{ll} 1/(2^{p'-1}-1), & 1<p<2, \\[1ex] 1, & p=2, \\[1ex] p'/\big(p'-1+\hat{\kappa}(p')\big), & p>2, \end{array} \right. \qquad \hat{\kappa}(r) = \dfrac{1+r(r-1)^{\frac{1}{r-2}}+(r-1)^{\frac{r}{r-2}}} {\big(1+(r-1)^{\frac{1}{r-2}}\big)^r}. \notag \end{align}\tag{22}\]
Theorem 2 (Existence). Let \(f \in L^1(0,1)\) and assume that \[\label{concomex95tvp} H^*+h_M \leq \Big(\kappa(p) \int_0^1 f(u)\, du \Big)^{1/p'}.\tag{23}\] Then there exists a unique \(c^* > H^*\) such that the problem 8 has a unique positive solution. Moreover, \(c^*\) satisfies \[\label{c9442} c^* \leq \max \Big\{ \frac{1}{\theta} \Big[ \Big( p'\int_0^1 f(u) \, du \Big)^{1/p'} - H(\theta) \Big], ~ -h_m \Big\}.\tag{24}\]
Remark 2. Our existence result is obtained through a TVP formulation as in [1]. However, if one adopts a forward initial value approach as in [2], then one may expect existence for \(c^* \geq H^*\) under the corresponding strict inequality assumption in 23 .
Before proving Theorem 2, we first establish the following lemma, which is a modification of [1] adapted to our setting.
Lemma 1. If \(f \in L^1(0,1)\) and 23 holds, then \[\hat{y}_{H^*}(\theta)^{1/p'}>H^*\theta+H(\theta).\]
Proof. To argue by contradiction, assume that \[\label{lemmaB462951} \hat{y}_{H^*}(\theta)^{1/p'} \leq H^*\theta+H(\theta).\tag{25}\] For a given \(\tau \in (\theta, 1)\), dividing 21 by \(y_c(u)^{1/p}\) and integrating it with \(c=H^*\) over \((\theta, \tau)\) gives \[\label{comtau95tvp} \begin{align} \hat{y}_{H^*}(\tau)^{1/p'} & =\hat{y}_{H^*}(\theta)^{1/p'} +\int_{\theta}^{\tau} (H^*+h(u)) \,du-\int_{\theta}^{\tau} \frac{f(u)}{\hat{y}_{H^*}(u)^{1/p}} \, du \\ & < \hat{y}_{H^*}(\theta)^{1/p'} + (H^*+h_M)(1-\theta). \end{align}\tag{26}\]
On the other hand, integrating 21 over \((\theta, 1)\) with \(c=H^*\) and applying the mean value theorem, there exists \(\tau^* \in (\theta, 1)\) such that \[\begin{align} 0 & \leq \hat{y}_{H^*}(\theta)+ p' (H^*+h_M) \int_{\theta}^1\hat{y}_{H^*}(u)^{1/p}\,du-p'\int_{\theta}^1 f(u) \, du \\ & = \hat{y}_{H^*}(\theta)+p'\hat{y}_{H^*}(\tau^*)^{1/p}(H^*+h_M)(1-\theta)-p'\int_{\theta}^1 f(u)\, du. \notag \end{align}\] It follows from 25 and 26 that \[\label{w295tvp} 0 < (H^*\theta+H(\theta))^{p'} +p' \big(H^*\theta+H(\theta) + (H^*+h_M)(1-\theta) \big)^{p'-1} (H^*+h_M)(1-\theta) -p'\int_{0}^1 f(u) \,du. \notag\tag{27}\] Here, we observe that both \(\alpha:=H^*\theta+H(\theta)\) and \(\beta:=(H^*+h_M)(1-\theta)\) are nonnegative. The above inequality has exactly the same form as (B.7) in [1], with \(H(\theta)\) and \(H(1)-H(\theta)\) replaced by \(\alpha\) and \(\beta\), respectively. Therefore, arguing exactly as below (B.7), one obtains a contradiction to 23 . The key difference from [1] is that \[0 \leq \alpha+\beta=\int_0^{\theta} (H^*+h(u))\,du+\int_{\theta}^1 (H^*+h_M)\,du \leq H^*+h_M, \notag\] which yields the left–hand side of 23 . In contrast, the quantity \(H(1)=H(\theta)+(H(1)-H(\theta))\) appears in [1]. ◻
We are now ready to prove Theorem 2.
The proof of Theorem 2. Following the strategy of [2], but using a bacward terminal value approach instead of the forward initial value formulation, we define two sets \[\mathcal{A}:=\{ c \geq H^* : u_c =0, ~ \hat{y}_c(0)>0\}, \quad \mathcal{B} = \{ c \geq H^*: u_c >0, ~\hat{y}_c(u_c)=0 \}. \notag\] Thanks to the continuity of \(\hat{y}_c\) and the continuous dependence of \(\hat{y}_c\) on \(c\), both \(\mathcal{A}\) and \(\mathcal{B}\) are open. Our strategy is to show that \(\sup \mathcal{A}=\inf \mathcal{B}\). We also note that if \(c \notin \mathcal{A}\cup\mathcal{B}\), then \(\hat{y}_c\) is a positive solution of 8 .
We first prove that both \(\mathcal{A}\) and \(\mathcal{B}\) are nonempty. Assume that \(\mathcal{B}=\emptyset\). Then, for every \(c\geq H^*\), we have \(u_c=0\) and \(\hat{y}_c(0)\geq0\). Dividing 21 by \(y_c(u)^{1/p}\) and integrating over \([0,\theta]\) yields \[\hat{y}_c(\theta)=\big(\hat{y}_c(0)^{1/p'}+c\theta+H(\theta)\big)^{p'}\geq (c\theta+H(\theta))^{p'}. \notag\] On the other hand, for \(c>-h_m\), integrating 21 over \([\theta,1]\) gives \[\hat{y}_c(\theta) = -p'\int_{\theta}^1 (c+h(u)) \hat{y}_c(u)^{1/p}\,du + p'\int_{\theta}^1 f(u)\,du < p'\int_{\theta}^1 f(u)\,du. \notag\] Consequently, \[(c\theta+H(\theta))^{p'} < p'\int_{\theta}^{1} f(u)\,du, \notag\] which is impossible for sufficiently large \(c\) satisfying \[\label{clarge} c > -h_m \quad \text{and} \quad c \geq \frac{1}{\theta} \left[ \left(p'\int_{\theta}^{1} f(u)\,du\right)^{1/p'} -H(\theta) \right].\tag{28}\] This contradiction proves that \(\mathcal{B}\neq\emptyset\).
We now claim \(H^* \in \mathcal{A}\) by proving \(\hat{y}_{H^*}(u_{H^*})>0\). Dividing 21 by \(y_c(u)^{1/p}\) and integrating over \([u_{H^*},\theta]\) gives \[\hat{y}_{H^*}(\theta)^{1/p'}-\hat{y}_{H^*}(u_{H^*})^{1/p'}=H^*(\theta-u_{H^*})+H(\theta)-H(u_{H^*}).\] Hence, by Lemma 1 and the fact that \(u_{H^*}\in[0,\theta]\), \[\hat{y}_{H^*}(u_{H^*})^{1/p'} > H^* u_{H^*}+H(u_{H^*}) \geq 0,\] which implies that \(u_{H^*}=0\) and \(H^* \in \mathcal{A}\). In particular, \(\mathcal{A}\neq\emptyset\).
Since \(\hat{y}_c\) is strictly decreasing with respect to \(c\), it follows that \(\sup \mathcal{A}\leq \inf \mathcal{B}\). Suppose that \(\sup \mathcal{A}<\inf \mathcal{B}\). Then there exist \(c_1<c_2\) such that \(c_1,c_2\notin\mathcal{A}\cup\mathcal{B}\). Hence, both \(\hat{y}_{c_1}\) and \(\hat{y}_{c_2}\) are positive solutions of 8 , and thus \(\hat{y}_{c_2}(u)<\hat{y}_{c_1}(u)\) on \((0,1)\). However, this contradicts 9 , since \[\hat{y}_{c_1}(u)=(c_1u+H(u))^{p'}<(c_2u+H(u))^{p'}=\hat{y}_{c_2}(u) \quad \text{on (0,\theta)}.\] Therefore, \(\sup \mathcal{A}=\inf \mathcal{B}\). Since \(\mathcal{A}\) and \(\mathcal{B}\) are open, the critical value \(c^*:=\sup \mathcal{A}=\inf \mathcal{B}\) satisfies \(c^*\notin\mathcal{A}\cup\mathcal{B}\). Hence, 8 has a unique positive solution \(y_c\) if and only if \(c=c^*\).
If \(c^*\) is sufficiently large so as to satisfy 28 , then necessarily \(c^*\in\mathcal{B}\) by the same argument used to prove that \(\mathcal{B}\neq\emptyset\). Since \(c^*\notin\mathcal{B}\), the estimate 24 follows. ◻
Example 2. The novelty of our existence result lies in showing the possibility of traveling waves with wave speeds in the range 7 . More precisely, the assumption 23 suggests that if the overall adverse convective effect remains sufficiently weaker than the propagation mechanism generated by the reaction–diffusion structure, then traveling waves can still persist even in regions where the convective effect acts against the propagation direction. Such a situation indeed occurs when \(f\) and \(h\) simultaneously satisfy the existence condition 23 and the nonexistence condition 20 from [1], the latter guaranteeing nonexistence for all \(c\ge -h_m\).
For example, this may occur when \(H^*+h_M\) is relatively small, while \(H(\theta)-\theta h_m\) is relatively large. Let \(p=p'=2\), \(\theta=\frac{1}{2}\), \(d(u) \equiv 1\), \(h(u)=-1-u\) on \([0,1]\), and \[g(u)= \begin{cases} 0, & 0\le u\le \frac{1}{2},\\[1mm] \frac{78}{25}\left(u-\frac{1}{2}\right)(1-u), & \frac{1}{2}<u<1. \end{cases} \notag\] Then \(h_m=-2\), \(h_M=-1\), and \(H^*=\frac{5}{4}<-h_m\). A direct computation shows that both 23 and 20 are satisfied. Consequently, the corresponding wave speed \(c^*\) necessarily satisfies 7 .