TY - JOUR
T1 - Existence of fast traveling waves for some parabolic equations
T2 - A dynamical systems approach
AU - Scheel, Arnd
PY - 1996/1/1
Y1 - 1996/1/1
N2 - We study semilinear elliptic equations Δu + cux=f(u,∇ u) and Δ2u + cux,= f(u, Δu, ∇2u) in infinite cylinders (x,-y)∈R×Ω⊂Rn+1 using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limit c→∞, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.
AB - We study semilinear elliptic equations Δu + cux=f(u,∇ u) and Δ2u + cux,= f(u, Δu, ∇2u) in infinite cylinders (x,-y)∈R×Ω⊂Rn+1 using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limit c→∞, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.
KW - Inertial manifolds
KW - Singular perturbation
KW - Traveling waves
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U2 - 10.1007/BF02218843
DO - 10.1007/BF02218843
M3 - Article
AN - SCOPUS:0001256341
SN - 1040-7294
VL - 8
SP - 469
EP - 547
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
IS - 4
ER -