How travelling waves attract the solutions of KPP-type equations
The Kolmogorov-Petrovskii-Piskunov (KPP) reaction diffusion equations NEWLINE\[NEWLINEu_t -\text{div}(A(x,y)Du) + B(x,y).Du =f(x,y,u), \quad (x,y)\in \mathbb{R}\times \mathbb{T}^{N-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{t\to -\infty} u(t,x,y) = 0, \lim_{t\to \infty} u(t,x,y) = 1,NEWLINE\]NEWLINE with \(f\) smooth, concave and 1-periodic in \(u\), with \(f(x,y,0)=f(x,y,1)=0\) and NEWLINE\[NEWLINE\sup_{(x,y)}f_u(x,y,1)<0<\sup_{(x,y)}f_u(x,y,0),NEWLINE\]NEWLINE and with \( \mathbb{T}^n\) being the \(n\)-dimensional torus, is known to have pulsating waves \(u(t,x,y)=\phi_c(t,x-ct,y)\) if \(c>c_*\) for some critical values \(c_*>0\). The authors study the large time asymptotics of solutions with initial data \(u_0\) lying between two translates of one pulsating wave: NEWLINE\[NEWLINE\phi_c(-M,x,y) \leq u_0(x,y) \leq \phi_c(M,x,y) \quad \text{for some} \;M>0, c>c_*.NEWLINE\]NEWLINE They show that the solution \(u\) converges to the pulsating wave with a shift \(m(t,x,y)\) determined as a solution of a nonlinear parabolic equation with periodic coefficients. In detail NEWLINE\[NEWLINE\sup_{(x,y)\in \mathbb{R}\times \mathbb{T}^{N-1}}|u(t,x,y)-\phi_c(t+m(t,x,y),x,y)|\to 0 \quad \text{as} \;t\to \infty.NEWLINE\]NEWLINE In the 1D case \(u_t-u_{xx}=f(u)\) with \(\phi_c(x-M)\leq u_0(x)\leq \phi_c(x+M)\) an explicit approximation of \(m\) is provided employing the Cole-Hopf transformation of the \(m\)-equation to a linear one and using the heat kernel. Moreover, in the 1D case a convergence rate for the asymptotics of \(u\) is obtained.NEWLINENEWLINEThe main techniques of the proof are the derivation of a simplified model for the shift \(m\), which can be analyzed, and linearizing it via the Cole-Hopf transform. For the convergence at \(t\to \infty\) the authors use mainly parabolic estimates and the weak maximum principle.
scientific article; zbMATH DE number 6191474
scientific article