TY - JOUR
T1 - An entire solution of a bistable parabolic equation on R with two colliding pulses
AU - Matano, H.
AU - Poláčik, P.
N1 - Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2017/3/1
Y1 - 2017/3/1
N2 - We consider semilinear parabolic equations of the form ut=uxx+f(u),x∈R,t∈I, where I=(0,∞) or I=(−∞,∞). Solutions defined for all (x,t)∈R2 are referred to as entire solutions. Assuming that f∈C1(R) is of a bistable type with stable constant steady states 0 and γ>0, we show the existence of an entire solution U(x,t) of the following form. For t≈−∞, U(⋅,t) has two humps, or, pulses, one near ∞, the other near −∞. As t increases, the humps move toward the origin x=0, eventually “colliding” and forming a one-hump final shape of the solution. With respect to the locally uniform convergence, the solution U(⋅,t) is a heteroclinic orbit connecting the (stable) steady state 0 to the (unstable) ground state of the equation uxx+f(u)=0. We find the solution U as the limit of a sequence of threshold solutions of the Cauchy problem for equation (A).
AB - We consider semilinear parabolic equations of the form ut=uxx+f(u),x∈R,t∈I, where I=(0,∞) or I=(−∞,∞). Solutions defined for all (x,t)∈R2 are referred to as entire solutions. Assuming that f∈C1(R) is of a bistable type with stable constant steady states 0 and γ>0, we show the existence of an entire solution U(x,t) of the following form. For t≈−∞, U(⋅,t) has two humps, or, pulses, one near ∞, the other near −∞. As t increases, the humps move toward the origin x=0, eventually “colliding” and forming a one-hump final shape of the solution. With respect to the locally uniform convergence, the solution U(⋅,t) is a heteroclinic orbit connecting the (stable) steady state 0 to the (unstable) ground state of the equation uxx+f(u)=0. We find the solution U as the limit of a sequence of threshold solutions of the Cauchy problem for equation (A).
KW - Entire solutions
KW - Parabolic equations on R
KW - Slowly moving pulses
KW - Threshold solutions
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U2 - 10.1016/j.jfa.2016.11.006
DO - 10.1016/j.jfa.2016.11.006
M3 - Article
AN - SCOPUS:85006802349
SN - 0022-1236
VL - 272
SP - 1956
EP - 1979
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 5
ER -