Variational Principles for Travelling Waves of
Reaction-diffusion Equations
Oct 19-24, 2009
Workshop on Kinetic Theory and Fluid Dynamics National Institute for Mathematical Sciences
and
Reaction-diffusion equations
letu be a vector-valued function,A be a diagonal
nonnegative matrix.
∂u
∂t = A∆u+f(u). (RD)
Reaction-diffusion equations describe phenomena in chemical kinetics, physics, and biology
such as combustion, phase transitions,
Travelling Waves:
( ∂u
∂t = ∆u+f(u) on R
u(x,t) =u(x−ct)
Min-max principles for the wave speed
Hadeler and Rothe (1975): K ={v∈C2
:vz<0,0<v <1}
ψ(v(z)) = ∆v(z) +f(v(z))
vz(z)
Theorem
sup v∈K
inf
Min-max principles for the wave speed
Hadeler and Rothe (1975): K ={v∈C2
:vz<0,0<v <1}
ψ(v(z)) = ∆v(z) +f(v(z))
vz(z)
Theorem
sup v∈K
inf
z∈Rψ(v(z)) =c = infv∈Ksupz∈Rψ(v(z)).
Min-max principles for the wave speed
Hadeler and Rothe (1975): K ={v∈C2
:vz<0,0<v <1}
ψ(v(z)) = ∆v(z) +f(v(z))
vz(z)
Theorem
sup v∈K
inf
z∈Rψ(v(z)) =c = infv∈Ksupz∈Rψ(v(z)).
◮ Vol’pert et al. (1994): monotone systems.
◮ Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
Min-max principles for the wave speed
Hadeler and Rothe (1975): K ={v∈C2
:vz<0,0<v <1}
ψ(v(z)) = ∆v(z) +f(v(z))
vz(z)
Theorem
sup v∈K
inf
z∈Rψ(v(z)) =c = infv∈Ksupz∈Rψ(v(z)).
◮ Vol’pert et al. (1994): monotone systems.
◮ Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
Min-max principles for the wave speed
◮ Vol’pert et al. (1994): monotone systems.
◮ Gartner-Freidlin (1979), Heinze-Papanicolaou-Stevens (2001):
inhomogeneous media.
◮ Assumptions: monotonicity of the wave, uniqueness, stability
◮ Question: Can we use this principle to find the travelling
Variational formulation for the solution
Variational problem: minv∈H1
Variational formulation for the solution
Variational problem: minv∈H1
c Ec(v)
Variational formulation for the solution
Variational problem: minv∈H1
c Ec(v)
◮ Question: Whichc should we choose?
◮ Translation in space:
Variational formulation for the solution
◮ Translation in space:
Variational formulation for the solution
Fife-McLeod (1977): cut-off ofEc to prove stability of travelling
wave
Ec as a variational problem: Heinze (2000), Muratov-Lucia-Novaga
(2004 -), Xinfu Chen (2006), Chen.
Theorem(Lucia-Muratov-Novaga, 2008)
Gradient system on a infinite cylinders: ∃ c∗ such that infEc∗= 0 and it can be attained by a travelling wave solution.
Proof: Constraint Z
ecz|∇v|2= 1.
Gallay-Risler (preprint): UseEc to prove the stability of travelling
Gradient system
Gradient system on a infinite cylinders with Neumann boundary condition:
∂u
∂t = A∆u− ∇uF(u). (1)
Theorem
(1)coercive: F(u)→ ∞ as|u| → ∞;
(2)w non-degenerate local min. point of F,F(w)>infF.
Lyapunov Fuction
(1) Infinitely many Lyapunov fuctions
Open Problems
(1) Different diffusion rates
