An entire solution of a bistable parabolic equation on \(\mathbb{R}\) with two colliding pulses
The authors are concerned with the study of the parabolic equation \[ u_t=u_{xx}+f(u), \quad x\in {\mathbb R}, \, t\in {\mathbb R}.\eqno{(*)} \] The function \(f\in C^1({\mathbb R})\) satisfies: there exist \(\gamma>\beta>0\) such that \(f(0)=f(\beta)=f(\gamma)=0\), \(f'(0)<0\), \(f'(\beta)>0\), \(f'(\gamma)<0\), \(f<0\) in \((0,\beta)\), \(f>0\) in \((\beta,\gamma)\) and \(\int_0^\gamma f(s)ds>0\). The main result of the paper reads as follows. There exists a solution \(U\) of (*) with the following properties: (i) \(0<U(\cdot,t)<\gamma\), \(U(\cdot,t)\in C_0({\mathbb R})\) and \(U(\cdot,t)\) is even, for all \(t\in {\mathbb R}\), (ii) For each \(t<0\), \(U(\cdot,t)\) has exactly three critical points, precisely two global maximum points and one local minimum point at \(x=0\). For all sufficiently large \(t>0\), \(x=0\) is the unique critical point of \(U(\cdot,t)\) and it is the global maximum point of \(U(\cdot,t)\), (iii) \(\lim_{t\to \infty}\|U(\cdot,t)-\phi\|_{L^\infty({\mathbb R})}=0\), where \(\phi\) satisfies \(\phi_{xx}+f(\phi)=0\) in \({\mathbb R}\) and \(\phi'(0)=0\), (iv) There exists a \(C^1\) function \(\zeta\in C^1(-\infty,0)\) such that \(\zeta(t)\to \infty\), \(\zeta'(t)\to 0\) as \(t\to -\infty\) and \[ \lim_{t\to -\infty}\|U(\cdot,t)-\phi(\cdot-\zeta(t))-\phi(\cdot+\zeta(t))\|_{L^\infty({\mathbb R})}=0. \]
scientific article; zbMATH DE number 6675456
scientific article