Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion

TL;DR

This paper develops a variational framework to analyze the asymptotic speed of reaction-diffusion fronts with nonlinear, gradient-dependent diffusion coefficients, providing bounds and exact speeds in special cases.

Contribution

It introduces a variational principle for front speed in reaction-diffusion equations with gradient-dependent diffusion, extending classical bounds to nonlinear, gradient-dependent cases.

Findings

Established bounds for front speed for all m ≥ 0, p ≥ 1.

Recovered classical bounds in the case m=1, p=2.

Determined exact front speed when m(p-1)=1.

Abstract

We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form $D(u,u_x) = m u^{m-1} u_x^{m(p-2)}$ for which existence and convergence to traveling fronts has been established. We formulate a variational principle for the asymptotic speed of the fronts. Upper and lower bounds for the speed valid for any $m\ge0, p\ge 1$ are constructed. When $m=1, p=2$ the problem reduces to the constant diffusion problem and the bounds correspond to the classic Zeldovich Frank-Kamenetskii lower bound and the Aronson-Weinberger upper bound respectively. In the special case $m(p-1) = 1$ a local lower bound can be constructed which coincides with the aforementioned upper bound. The speed in this case is completely determined in agreement with recent results.

Equations

$${\partial }_{t}u={\partial }_x(|{\partial }_x{u}^m{|}^{p-2}{\partial }_x{u}^m)+f(u),f(0)=f(1)=0,f(u)>0\text{mbox}in(0,1),$$

$$c_{\ast }(m,p){|}_{\gamma =0}\equiv c_0(m,p)=(m^2{p}^{m+1}{f}^{\prime }(0){)}^{1/(m+1)}.$$

$$c_{\ast }=\underset{g}{\sup }\left[p{\left(\frac{m}{p-1}\right)}^{(p-1)/p}\frac{{\int }_0^1{u}^{(m-1)(p-1)/p}{h}^{1/p}{f}^{(p-1)/p}{g}^{(p-1)/p}du}{{\int }_0^{1}g(u)du}\right],$$

$${\left(\frac{mp}{p-1}{\int }_0^1{u}^{(m-1)}f(u)du\right)}^{(p-1)/p}\le c_{\ast }(m,p)\le$$

$$p{\left(\frac{m}{p-1}\right)}^{(p-1)/p}\underset{u}{\sup }{\left[u^{\gamma }{\left(\frac{f}{u}\right)}^{(p-1)}\right]}^{1/p}.$$

$$c_{\ast }\ge c_{ZFK}=\sqrt{2{\int }_0^{1}f(u)du}$$

$$c\le \underset{u}{\sup }2\sqrt{\frac{f(u)}{u}}.$$

$$c{u}_{\xi }=m^{p-1}\frac{d}{d\xi }\left(u^{(m-1)(p-1)}(u_{\xi }{)}^{p-1}\right)+f(u).$$

$$q(u)=u^{m-1}{u}^{\prime }(u)$$

$$\frac{c}{m^{p-1}}=\frac{d}{du}(q(u){)}^{p-1}+\frac{u^{m-1}f(u)}{m^{p-1}q(u)}.$$

$$F(u)=\frac{u^{m-1}f(u)}{m^{p-1}}.$$

$$\mathcal{J}[g]\equiv \frac{p{m}^{p-1}}{(p-1{)}^{(p-1)/p}}\frac{{\int }_0^{1}h(u{)}^{1/p}F(u{)}^{(p-1)/p}{g}^{(p-1)/p}du}{{\int }_0^{1}g(u)du},$$

$$c_{*}(m,p)=J\equiv\sup\{\mathcal{J}[g]\bigm{|}g\in\mathcal{D}\}$$

$$\frac{c}{m^{p-1}}{\int }_0^{1}g(u)du={\int }_0^{1}du\left(h(u)q(u{)}^{p-1}+\frac{g(u)F(u)}{q(u)}\right)\equiv {\int }_0^1\Phi (u)du.$$

$$\Phi =h(u)q(u{)}^{p-1}+\frac{g(u)F(u)}{q(u)},$$

$$\hat{q}={\left[\frac{Fg}{(p-1)h}\right]}^{1/p},$$

$$\Phi (\hat{q})=\frac{pg{h}^{1/p}{F}^{(p-1)/p}}{(p-1{)}^{(p-1)/p}}.$$

$$c_{\ast }\ge \frac{p{m}^{p-1}}{(p-1{)}^{(p-1)/p}}\frac{{\int }_0^1{h}^{1/p}{F}^{(p-1)/p}{g}^{(p-1)/p}du}{{\int }_0^{1}g(u)du},$$

$$c_{\ast }=\underset{g}{\sup }\frac{p{m}^{p-1}}{(p-1{)}^{(p-1)/p}}\frac{{\int }_0^1{h}^{1/p}{F}^{(p-1)/p}{g}^{(p-1)/p}du}{{\int }_0^{1}g(u)du}.$$

$$c_{\ast }=\underset{g\in \mathcal{D}}{\max }(\mathcal{J}[g]).$$

$$q(u{)}^{p-1}\approx \frac{c_{\ast }}{m^{p-1}}u,$$

$$\frac{v^{\prime }}{v}=\frac{c_{\ast }}{m^{p-1}}\frac{1}{q^{p-1}(u)},$$

$$\frac{v^{\prime \prime }}{v}=\frac{c_{\ast }}{m^{p-1}}\frac{F(u)}{q^{2p-1}(u)}.$$

$$\tilde{g}(u)=\frac{1}{(v^{\prime }(u){)}^{1/(p-1)}},$$

$$-{\tilde{g}}^{\prime }(u)=\frac{1}{p-1}\frac{1}{(v^{\prime }(u){)}^{p/(p-1)}}{v}^{\prime \prime }=\frac{1}{p-1}\frac{1}{(v^{\prime }(u){)}^{/(p-1)}}\frac{v^{\prime \prime }}{v}\frac{v}{v^{\prime }}$$

$$=\frac{1}{p-1}g(u)\frac{F(u)}{q^p(u)}.$$

$$v(u)\approx Au\text{mbox}and{v}^{\prime }(u)\approx A,$$

$$\tilde{g}(0)=A^{-1/(p-1)}<\infty .$$

$$v(u)=\exp \left({\int }_{u_0}^u\frac{c_{\ast }}{m^{p-1}}\frac{1}{q^{p-1}(s)}ds\right),$$

$$\tilde{g}(u)=\frac{mq(u)}{{c_{\ast }}^{1/(p-1)}}\exp \left(\frac{1}{p-1}{\int }_u^{u_0}\frac{c_{\ast }}{m^{p-1}}\frac{1}{q^{p-1}(s)}ds\right).$$

$$g_{\alpha }(u)=\alpha g_1(u)+(1-\alpha )g_2(u).$$

Full text

1   Introduction

In this work we study the asymptotic propagation of fronts of the scalar reaction diffusion equation,

[TABLE]

which reduces to the classical problem KPP37 when $m=1,p=2$. The diffusion term can be seen either as the scalar version of the p-Laplacian acting on $u^{m}$ or as reaction diffusion equation with nonlinear diffusion coefficient $D(u,u_{x})=mu^{m-1}|u_{x}|^{m(p-2)}$. Such diffusion coefficients are encountered, for example, in hot plasmas Jardin2008 ; Wilhelm2001 and the corresponding processes are referred to as doubly nonlinear diffusion processes Vasquez .

The classical problem $m=1$, $p=2$, is fully understood AW78 ; KPP37 . When nonlinear diffusion is included several scenarios may arise depending on the precise form of the diffusion coefficient. The case of a power of concentration diffusion coefficient of the form $D(u)=u^{s}$ has been studied extensively beginning with the analytical solution found for $s=2$. Existence and convergence results are known for all $s$. A distinctive feature of density dependent diffusion is the appearance of a finite wave at the asymptotic speed. This is true even in the simpler $p=2$ case when $m>1$ (see, e.g., Ar80 ; BeDe95 , and references therein). In recent work Vasquez the more general case of doubly nonlinear diffusion is considered. It is shown that for all $m>0,p>1$ such that $\gamma=m(p-1)-1>0$, a unique monotonic increasing travelling wave joining the equilibria $u=0$ and $u=1$ exists for speeds $c\geq c_{*}(m,p)$ and none if $0<c<c_{*}(m,p)$. For $c=c_{*}(m,p)$ the travelling wave (TW) is finite, whereas for $c>c_{*}(m,p)$ the TW is positive (see Vasquez , Theorem 2.1). In the case $\gamma=0$ a unique monotonic increasing travelling wave joining the equilibria $u=0$ and $u=1$ exists for speeds $c\geq c_{*}(m,p)$ and none if $0<c<c_{*}(m,p)$. For $c=c_{*}(m,p)$ the travelling wave (TW) is positive (see Vasquez , Theorem 2.2). Moreover, when $\gamma=0$, an explicit expression for the minimal speed is given,

[TABLE]

The convergence of suitable initial conditions to the travelling wave of minimal speed is demonstrated in Vasquez as well.

The purpose of this work is to establish a variational characterization for the speed $c_{*}(m,p)$. The exact value of the speed cannot be determined in general however upper and lower bounds on the speed for general values of $m$ and $p$ can be obtained. The main result of the present work (see Theorem 1 below) is the variational expression for the speed

[TABLE]

from where upper and lower bounds will be constructed. In (3), $g\in C^{1}(0,1)$ is such that $g(u)\geq 0$, with $h(u)\equiv-g^{\prime}(u)>0$ in $(0,1)$ and $\int_{0}^{1}g(u)\,du$ finite.

We find that for any $m>0,p>1$ (with $\gamma\geq 0$) the asymptotic speed is bounded by

[TABLE]

The lower bound is a generalization of the Zeldovich-Frank-Kamenetskii (ZFK) bound (see, e.g., BeNi92 ; BeDe98 ). Effectively, for $m=1,p=2$ their classical bound

[TABLE]

is recovered. The upper bound, for $m=1,p=2$ reduces to the Aronson-Weinberger upper bound AW78

[TABLE]

An interesting case arises when $\gamma=0$. As mentioned above the speed can be determined exactly Vasquez and it is given by $c_{0}(m,p)$ when $\gamma=0$. Here we recover this result from the variational principle showing that when $\gamma=0$ a local lower bound can be found choosing an adequate trial function $g(u)$. This lower bound is exactly $c_{0}(m,p)$. The upper bound given in (4) reduces to $c_{0}(m,p)$ when $\gamma=0$ and $f(u)$ satisfies the KPP criterion $sup_{u}\sqrt{f(u)/u}=f^{\prime}(0).$ In the following sections we prove the statements made above. Our variational principle reduces to our standard variational principle (see BeDe96CMP , BeDe96PRL , BeDe98 ) when $p=2$.

The rest of this manuscript is organized as follows: In Section 2 we derive the variational principle, in Section 3 the bounds for general values of $\gamma$ with $m>0$, $p>1$ are obtained, and in Section 4 we derive a lower bound of the Zeldovich-Frank-Kamenetskii type for any $\gamma\geq 0$.

2   Variational Principle

We consider left travelling wave solutions $u(\xi)$ with $\xi=x+ct$ so that the TW profile satisfies $u_{\xi}>0.$ The TW solution satisfies the ordinary differential equation (ODE)

[TABLE]

From here on we denote $u^{\prime}=u_{\xi}$. Following the usual procedure, we introduce the phase space coordinate

[TABLE]

in terms of which the ODE for the travelling waves becomes, after dividing by $q,$

[TABLE]

Here, it is convenient to define

[TABLE]

In what follows, let us define the functional

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which acts on $\mathcal{D}$, the space of functions $g\in C^{1}(0,1)$ such that $g(u)\geq 0$, with $h(u)=-g^{\prime}(u)>0$ in $(0,1)$ and $\int_{0}^{1}g(u)\,du$ finite. Here the function $F(u)$ is given by (7) above. With this notation we state our main result, which is embodied in the following theorem.

Theorem 2.1 (Variational characterization of $c_{*}$)

Let $f\in C^{1}[0,1]$ with $f(0)=f(1)=0$, $f(u)>0$ in $(0,1)$, and $f(u)$ concave in $[0,1]$. Assume $\gamma=m(p-1)-1\geq 0$. Then,

[TABLE]

Moreover,

i) If $\gamma>0$, there is a $g\in\mathcal{D}$, $\tilde{g}$ say, such that $J=\mathcal{J}[\tilde{g}]$. This maximizing $\tilde{g}$ is unique up to a multiplicative constant, and

ii) If $\gamma=0$ we construct and explicit maximizing sequence $g_{\alpha}\in\mathcal{D}$ such that $\lim_{\alpha\to 0}\mathcal{J}[g_{\alpha}]=c_{*}(m,p)|_{\gamma=0}$, where $c_{*}(m,p)|_{\gamma=0}$ is given by (2) above.

Proof

Let $g(u)\in\mathcal{D}$. Multiplying (6) by $g(u)$ and integrating in $u$ between [math] and $1$ we obtain after integrating by parts,

[TABLE]

where $h(u)\equiv-g^{\prime}(u)>0$ and we assume that $g(u)$ is such that

$\lim_{u\rightarrow 0}g(u)q(u)^{p-1}=0$.

The integrand of the right side,

[TABLE]

at fixed $u$ can be considered as a function of $q$. It is clear from (11) that $\Phi(q)$ has a unique positive minimum at $\hat{q}$ so that $\Phi(q)\geq\Phi(\hat{q})$. A simple calculation yields

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and

[TABLE]

It follows from (10) that

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for every $g\in\mathcal{D}$. To establish (9) we need only prove that the supremum of the right side of (13) over all $g\in\mathcal{D}$ is actually $c_{*}$. we will do this separately in the cases $\gamma>0$ and $\gamma=0$.

i) Case $\gamma>0$. Below we show that when $\hat{q}$ is the solution of (6) (with $c=c_{*}$), equality is attained in (13) for some $g\in\mathcal{D}$, so that we obtain the variational characterization for the speed

[TABLE]

We have already proven (see (13) above) that $c_{*}\geq\mathcal{J}[g]$ for every $g\in\mathcal{D}$. What we will actually show here is that when $\gamma>0$, there exists a $g\in\mathcal{D}$, $\tilde{g}$ say, such that $c_{*}=\mathcal{J}[\tilde{g}]$. Hence in the case $\gamma>0$ the variational principle reads,

[TABLE]

In the case $\gamma>0$, the existence of a travelling wave for any $c\geq c_{*}$ was proven in Theorem 2.1 of Reference Vasquez . Moreover, in the case $\gamma>0$, the solution of (6) satisfies,

[TABLE]

in the neighborhood of $u=0$. In order to show that the $\sup$ is actually attained in (9) we have to show that there exists $\tilde{g}\in\mathcal{D}$ satisfying (12) when $\hat{q}$ is a solution of (6). To construct such a $g$, let $v$ be the solution of

[TABLE]

where $q$ is a solution of (6). Notice that this $v$ is unique up to a multiplicative constant. A simple calculation using (17), (6), and the definition (7) of $F$, yields,

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Choosing

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it follows from (17) and (18) that,

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which is precisely (12). From (16) and (17) we have that

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near $u=0$. Hence, it follows from (19) that

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Integrating (17) and using (21) we can write explicitly,

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for some $0<u_{0}<1$. Clearly, the value of $A$ in (21) is determined by the value of $u_{0}$. Finally, using (17), (19), and (23), we can write,

[TABLE]

Since, the integrand in (24) is positive, $u_{0}<1$, and $q(1)=0$, it follows from (24) that $\tilde{g}(1)=0$. From all the results above it follows that $\tilde{g}$ given by (24) is in $\mathcal{D}$, and that $c_{*}=\mathcal{J}[\tilde{g}]$.

It is clear from the construction above that $\tilde{g}$ is unique up to a multiplicative cosntant. The uniqueness of the maximizing $g\in\mathcal{D}$, however, can be seen directly from our variational principle (8). In fact, suppose that there are two different maximizers, say $g_{1},g_{2}\in\mathcal{D}$, with $\int_{0}^{1}g_{1}(u)\,du=\int_{0}^{1}g_{2}(u)\,du=1$. Then, for any $\alpha\in(0,1)$ consider now,

[TABLE]

It is clear from (25) that $g_{\alpha}\in\mathcal{D}$ and that $\int_{0}^{1}g_{\alpha}(u)\,du=1$. Using Hölder’s inequality with exponents $p$ and $p^{\prime}=p/(p-1)$, it follows from (8) that

[TABLE]

which is a contradiction with the fact that $g_{1}$ and $g_{2}$ are the maximizers. Notice that the inequality in (26) is strict if $g_{1}\not\equiv g_{2}$.

ii) Case $\gamma=0$. For later purposes it is convenient to denote

[TABLE]

It then follows from (7) and (8) that

[TABLE]

in the case $\gamma=0$, when we conveniently normalize $g$ so that $\int_{0}^{1}g(u)\,du=1$. Now, choose as a trial function the sequence

[TABLE]

Notice that for each $\alpha\in(0,1)$, $g_{\alpha}(u)>0$, $g_{\alpha}^{\prime}(u)<0$, $g_{\alpha}(1)=0$, and $\lim_{u\to 0}[u\,g_{\alpha}(u)]=0$, so these are appropriate trial functions. Moreover, we have normalized the $g_{\alpha}$’s so that $\int_{0}^{1}g_{\alpha}(u)\,du=1$.

With this choice we will show that $\lim_{\alpha\rightarrow 0}J_{g_{\alpha}}[f]=f^{\prime}(0)^{1/(m+1)}$ so that $\mathcal{J}[g_{\alpha}]\to(m^{2}p^{m+1}f^{\prime}(0))^{1/(m+1)}=c_{0}(m,p)$ as $\alpha\to 0$. To do so we write $J[f]=J[uf^{\prime}(0)]+J[f]-J[uf^{\prime}(0)]$ and show that

[TABLE]

While the proof of the second limit is given in the Appendix, the proof of the first is as follows. Using (27) with $g=g_{\alpha}$ we have,

[TABLE]

where $B(x,y)$ denotes the Euler Beta function. Now, $B(t,s)=\Gamma(t)\Gamma(s)/\Gamma(t+s)$, hence

[TABLE]

Using $\lim_{x\to 0}x\,\Gamma(x)=1$ to evaluate the limit of the right side of (32) when $\alpha\to 0$, we finally conclude, $J_{g_{\alpha}}[uf^{\prime}(0)]\to f^{\prime}(0)^{1/(m+1)}$ as $\alpha\to 0$ from above. As indicated before, (30) then implies that $\mathcal{J}[g_{\alpha}]\to(m^{2}p^{m+1}f^{\prime}(0))^{1/(m+1)}=c_{0}(m,p)$ as $\alpha\to 0$, which concludes the proof of the Theorem.

3 An upper bound on the speed for $\gamma\geq 0$

In this section we derive from our variational principle (i.e., from Theorem 1 above) an explicit upper bound on the speed of fronts. In order to do this we rewrite (14) as

[TABLE]

Since the mapping $x\to x^{1/p}$ is concave for $p>1$, defining the probability measure $d\nu=g(u)du/\int_{0}^{1}gdu$, and using Jensen’s inequality we get,

[TABLE]

Integrating $\int_{0}^{1}h(u)\,u\,du$ by parts, using $g(1)=0$, and $\lim_{u\rightarrow 0}ug(u)=0$ it follows from (3) that

[TABLE]

Replacing the expression for $F(u)$ in terms of $f(u)$ we finally obtain the upper bound

[TABLE]

with $\gamma=m(p-1)-1$ as defined before. When $\gamma=0$ the expression above reduces to

[TABLE]

In particular, when $m=1$ (i.e., $p=2$ since $\gamma=0$), (36) is the classical upper bound of Aronson and Weinberger AW78 . Notice that for the reaction profiles considered here (i.e., $f(u)$ positive and concave in $[0,1]$, $f(0)=f(1)=0$ and $f\in C^{1}[0,1]$, we clearly have that $\sup_{u\in[0,1]}f(u)/u=f^{\prime}(0)$, and in fact we have equality in (36).

4   Integral lower bound: a Zeldovich–Frank–Kamenetskii type bound

From the variational characterization lower bounds can be constructed choosing specific values for the trial function $g(u)$. In this section we construct a lower bound which involves the integrals of the reaction term as the Zeldovich–Frank–Kamenetskii classical bound ZFK38 ; BeNi92 ; BeDe98 . Our ZFK type bound is embodied in the following lemma.

Lemma 1

For any $m>0$, $p>1$, $\gamma\geq 0$ and $f$ satisfying the hypothesis of Theorem 1, we have that

[TABLE]

Proof

Choose as a trial function of our variational principle (9) the function

[TABLE]

It is simple to verify that $g\in C^{1}(0,1)$, $h=-g^{\prime}>0$, $g(1)=0$, and $g(u)\geq 0$ in $[0,1]$. Moreover, since $g$ is decreasing, $\int_{0}^{1}g(u)du\leq g(0)=\left(\int_{0}^{1}F(u^{\prime})du^{\prime}\right)^{1/p}<\infty$. Hence, $g\in\mathcal{D}$. A simple calculation yields

[TABLE]

and $hg^{p-1}=F(u)/p$. It follows then from (14) that

[TABLE]

Now, since $g(u)$ is a decreasing positive function, $\int_{0}^{1}g(u)du\leq g(0)$. Hence,

[TABLE]

If we express the right side of (38) in terms of the original reaction term $f(u)$ we get (37) which proves the lemma.

Acknowledgements.

This work was supported by Fondecyt (Chile) projects 114–1155, 116–0856 and by Iniciativa Científica Milenio, ICM (Chile), through the Millennium Nucleus RC–120002.

Appendix

In this appendix we show that

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where $g_{\alpha}$ is given by (29). We defined

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so that

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Since for $m\geq 0$, $1/(m+1)\leq 1$, it is not difficult to verify the inequality $|a^{1/(m+1)}-b^{1/(m+1)}|\leq|a-b|^{1/(m+1)}$ for all $a\geq 0,b\geq 0,m\geq 0.$ In the present case, we have

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If $f(u)$ and its derivative are continuous in $[0,1]$, there exist $d>0,k>0$ such that

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Using (41) and (42) in (40), together with the explicit form of $g_{\alpha}$ we have that

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where

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Since $\alpha>0$ and $k>0$, $N(\alpha)>-1$ and $u^{N(\alpha)}$ is integrable. Performing the integral we finally find

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