The application of Hamilton-Jacobi equation in reaction-diffusion equations.

Abstract. In this article, we introduce the application of Hamilton-Jacobi equation in reaction-diffusion equation, which can be uesd to study the long time behavior of the R-D equation, which is an important project in biology and ecology.

1. Introduction.

\,\,\,\,Hamilton-Jacobi equation is an interesting branch in PDE, since 1980s, this tools has been used to study the long time behavior of the raction-iffusion equations. Precisely, this method combines the evolution equations and fully nonlinearly equations. and the method of geometric optics is based on deriving the limiting problem for large space and large time, for which, corresponding to HamiltonJacobi equations, the solution has to be understood in the viscosity sense. It was introduced by Freidlin{friedlin}, who employed probabilistic arguments to study the asymptotic behavior of solution to the Fisher-KPP equation modeling the population of a single species. Subsequently, the result was generalized by Evans and Souganidis using PDE arguments. The method was also applied by Barles, Evans and Souganidis [4] to study KPP systems, where several species spread at a common spreading speed. Here we introduce this optics briefly as follow.

2. Main results.

\,\,\,\,Considering the classical reaction-diffusion equation
(1) u t ( t , x ) = Δ u ( t , x ) + f ( u ( t , x ) ) t > 0 , x R N , (1) u t ( t , x ) = Δ u ( t , x ) + f ( u ( t , x ) ) t > 0 , x R N , {:(1)u_(t)(t","x)=Delta u(t","x)+f(u(t","x))quad t > 0","x inR^(N)",":}\begin{equation} u_t(t,x)=\Delta u (t,x)+f(u(t,x))\quad t>0, x\in\mathbb{R}^N, \end{equation}
here f f ff is Fisher-KPP nonlinearity, that is f ( 0 ) = f ( 1 ) = 0 f ( 0 ) = f ( 1 ) = 0 f(0)=f(1)=0f(0)=f(1)=0; f ( u ) > 0 f ( u ) > 0 f(u) > 0f(u)>0, f ( u ) f ( 0 ) u u ( 0 , 1 ) f ( u ) f ( 0 ) u u ( 0 , 1 ) f(u)leqslantf^(')(0)uAA u in(0,1)f(u)\leqslant f'(0)u~~\forall u\in(0,1); [1] equipped with compactly supported initial data u 0 ( x ) u 0 ( x ) u_(0)(x)u_0(x) with G 0 G 0 G_(0)G_0.
The hyperscaling of u u uu is given by
(2) u ε ( x , t ) = u ( t ε , x ε ) . (2) u ε ( x , t ) = u t ε , x ε . {:(2)u^( epsi)(x","t)=u((t)/( epsi),(x)/( epsi)).:}\begin{equation} u^\varepsilon(x,t)=u\left(\frac{t}{\varepsilon},\frac{x}{\varepsilon}\right). \end{equation}
Then [1] reduces to
(3) { u t ε ( x , t ) = ε Δ u ε ( x , t ) + 1 ε f ( u ε ( x , t ) ) , t > 0 , x R N , u ε ( x , 0 ) = u 0 ( x ε ) . (3) u t ε ( x , t ) = ε Δ u ε ( x , t ) + 1 ε f ( u ε ( x , t ) ) , t > 0 , x R N , u ε ( x , 0 ) = u 0 x ε . {:(3){[u_(t)^( epsi)(x","t)=epsiDeltau^( epsi)(x","t)+(1)/(epsi)f(u^( epsi)(x","t))","quad t > 0","x inR^(N)","],[u^( epsi)(x","0)=u_(0)((x)/( epsi)).]:}:}\begin{equation} \begin{cases} u^\varepsilon_ t(x,t)={\varepsilon} \Delta u^\varepsilon(x,t)+\displaystyle\frac{1}{\varepsilon}f(u^\varepsilon(x,t)), \quad t>0, x\in\mathbb{R}^N,\\ u^\varepsilon(x,0)=u_{0}\left(\frac{x}{\varepsilon}\right). \end{cases} \end{equation}
A Hopf-Cole transform is introduced
(4) w ϵ ( t , x ) = ϵ log u ϵ ( t , x ) , (4) w ϵ t , x = ϵ log u ϵ ( t , x ) , {:(4)w^(epsilon)(t,x)=-epsilon log u^(epsilon)(t","x)",":}\begin{equation} w^{\epsilon}\left( t,x \right) =-\epsilon\log u^{\epsilon}(t,x), \end{equation}
and we can verify that w ϵ w ϵ w^(epsilon)w^{\epsilon} satisfies
(5) { w t ϵ + | w ϵ | 2 + f ( u ϵ ) u ϵ = ϵ Δ w ϵ , t > 0 , x R N , w 0 ( x ) = { 0 in G 0 , + in R G ¯ 0 . (5) w t ϵ + | w ϵ | 2 + f ( u ϵ ) u ϵ = ϵ Δ w ϵ , t > 0 , x R N , w 0 ( x ) = 0 in G 0 , + in R G ¯ 0 . {:(5){[w_(t)^(epsilon)+|gradw^(epsilon)|^(2)+(f(u^(epsilon)))/(u^(epsilon))=epsilon Deltaw^(epsilon)",",t > 0","x inR^(N)","],[w_(0)(x)={[0,"in"quadG_(0)","],[+oo,"in"quadR\\ bar(G)_(0).]:}]:}:}\begin{equation} \begin{cases} w_{t}^{\epsilon}+| \nabla w^{\epsilon}|^{2}+\frac{f(u^{\epsilon})}{u^{\epsilon}}=\epsilon \Delta w^{\epsilon}, &t>0, x\in\mathbb{R}^N, \\ w_{0}(x)=\begin{cases} 0& \text{in}\quad G_{0},\\ +\infty& \text{in}\quad \mathbb{R}\backslash \bar G_{0}. \end{cases} \end{cases} \end{equation}
As ϵ + ϵ + epsilon rarr+oo\epsilon\rightarrow +\infty, [5] converges to the following Hamilton-Jacobi equation in the viscosity sense
(6) { t ϕ + | ϕ | 2 + f ( 0 ) = 0 , t > 0 , x R N , ϕ ( 0 , x ) = { 0 in G 0 , + in R G ¯ 0 . (6) t ϕ + | ϕ | 2 + f ( 0 ) = 0 , t > 0 , x R N , ϕ ( 0 , x ) = 0 in G 0 , + in R G ¯ 0 . {:(6){[del_(t)phi+|grad phi|^(2)+f^(')(0)=0",",t > 0","x inR^(N)","],[phi(0","x)={[0,"in"quadG_(0)","],[+oo,"in"quadR\\ bar(G)_(0).]:}]:}:}\begin{equation} \begin{cases} \partial _{t}\phi+| \nabla \phi|^{2}+f'(0)=0, &t>0, x\in\mathbb{R}^N, \\ \phi(0,x)=\begin{cases} 0& \text{in}\quad G_{0},\\ +\infty& \text{in}\quad \mathbb{R}\backslash \bar G_{0}. \end{cases} \end{cases} \end{equation}
As a result, it holds that
(7) u ϵ { 1 locally uniformly in Int { ϕ = 0 } , 0 locally uniformly in { ϕ > 0 } . (7) u ϵ 1 locally uniformly in Int { ϕ = 0 } , 0 locally uniformly in { ϕ > 0 } . {:(7)u^( epsilon)rarr{[1,"locally uniformly in"quad"Int"{phi=0}","],[0,"locally uniformly in"quad{phi > 0}.]:}:}\begin{equation} u^\epsilon\rightarrow \begin{cases} 1 & \text{locally uniformly in} \quad\textbf{Int}\{\phi=0\},\\ 0 & \text{locally uniformly in} \quad\{\phi>0\}. \end{cases} \end{equation}
Freidlin's condition. Freidlin's analysis of the asymptotics of u ε u ε u^(epsi)u^{\varepsilon} depends upon an auxiliary hypothesis, described as follows. For ( x , t ) R n × ( 0 , ) ( x , t ) R n × ( 0 , ) (x,t)inR^(n)xx(0,oo)(x, t) \in \mathbf{R}^{n} \times(0, \infty), write
J ( x , t ) inf x ( ) X { 0 t [ x ˙ i ( s ) x ˙ j ( s ) f ( 0 ) ] d s x ( 0 ) = x , x ( t ) G 0 } J ( x , t ) inf x ( ) X 0 t x ˙ i ( s ) x ˙ j ( s ) f ( 0 ) d s x ( 0 ) = x , x ( t ) G 0 {:[J(x","t)-=i n f_(x(*)in X){int_(0)^(t)[x^(˙)_(i)(s)x^(˙)_(j)(s)-f^(')(0)]ds∣:}],[{:x(0)=x,x(t)inG_(0)}]:}\begin{array}{r} J(x, t) \equiv \inf _{x(\cdot) \in X}\left\{\int_{0}^{t}\left[\dot{x}_{i}(s) \dot{x}_{j}(s)-f'(0)\right] d s \mid\right. \\ \left.x(0)=x, x(t) \in G_{0}\right\} \end{array}
and set P { J > 0 } P { J > 0 } P-={J > 0}P \equiv\{J>0\}. We say that Freidlin's condition holds provided that
J ( x , t ) = inf x ( ) X { 0 t [ x ˙ i ( s ) x ˙ j ( s ) f ( 0 ) ] d s x ( 0 ) = x , x ( t ) G 0 , ( x ( s ) , t s ) P for 0 < s t } for each ( x , t ) P . J ( x , t ) = inf x ( ) X 0 t x ˙ i ( s ) x ˙ j ( s ) f ( 0 ) d s x ( 0 ) = x , x ( t ) G 0 , ( x ( s ) , t s ) P  for  0 < s t  for each  ( x , t ) P . ∣{:[J(x","t)=i n f_(x(*)in X){int_(0)^(t)[x^(˙)_(i)(s)x^(˙)_(j)(s)-f^(')(0)]ds∣:}],[{:x(0)=x,x(t)inG_(0),(x(s),t-s)in P quad" for "0 < s <= t}],[" for each "(x","t)in del P.]:}\mid \begin{aligned} &J(x, t)=\inf _{x(\cdot) \in X}\left\{\int_{0}^{t}\left[\dot{x}_{i}(s) \dot{x}_{j}(s)-f'(0)\right] d s \mid\right. \\ &\left.x(0)=x, x(t) \in G_{0},(x(s), t-s) \in P \quad \text { for } 0<s \leq t\right\} \\ &\text { for each }(x, t) \in \partial P. \end{aligned}
As a result, we know from {friedlin-1985}, it holds that
Theorem. Assume that Friedlin's condition holds.Then
ϕ ( x , t ) = max { J ( x , t ) , 0 } ( x R n , t > 0 ) . ϕ ( x , t ) = max { J ( x , t ) , 0 } x R n , t > 0 . phi(x,t)=max{J(x,t),0}quad(x inR^(n),t > 0).\phi(x, t)=\max \{J(x, t), 0\} \quad\left(x \in \mathbf{R}^{n}, t>0\right).
Moreover, it can be solved explicitly as
lim ε 0 u ε ( x , t ) = { 0 if d ( x , G 0 ) > t 2 f ( 0 ) , 1 if d ( x , G 0 ) < t 2 f ( 0 ) . lim ε 0 u ε ( x , t ) = 0       if  d x , G 0 > t 2 f ( 0 ) , 1       if  d x , G 0 < t 2 f ( 0 ) . lim_(epsi rarr0)u^(epsi)(x,t)={[0," if "d(x,G_(0)) > t*2sqrt(f^(')(0))","],[1," if "d(x,G_(0)) < t*2sqrt(f^(')(0)).]:}\lim _{\varepsilon \rightarrow 0} u^{\varepsilon}(x, t)= \begin{cases}0 & \text { if } d\left(x, G_{0}\right)>t\cdot 2\sqrt{f'(0)}, \\ 1 & \text { if } d\left(x, G_{0}\right)<t\cdot 2\sqrt{f'(0)} .\end{cases}
We note that the interface propgates with the critical traveling wave speed 2 f ( 0 ) 2 f ( 0 ) 2sqrt(f^(')(0))2\sqrt{f'(0)}, Moreover, one can see {alfaro2011,alfaro2012} that the decay of the initial value has an influences on the progating speed of interface.

3. References.

  1. M. Alfaro, A. Ducrot, Sharp interface limit of the Fisher-KPP equation, Commun. Pure Appl. Anal.11 (2012) 1-18.
  2. M. Alfaro, A. Ducrot, Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011) 15-29.
  3. G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-di usion systems of PDE, Duke Math. J., 61 (1990), 835-858.
  4. L.C. Evans, P.E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J. 38 (1989) 141-172.
  5. M. Freidlin, Limit theorems for large deviations and reaction-diffusion equation, Ann. Probab.,13 (1985), 639-675.

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