Bifurcation of nonplanar travelling waves in a free boundary problem (Q5934269)

The stability of the planar travelling-wave solutions to a free-boundary problem arising from combustion theory with jump conditions at the interface defined by \(\zeta=\xi(y,t)\): NEWLINE\[NEWLINEu_t(t,\zeta,y)=\Delta u(t,\zeta,y)+ u(t,\zeta,y) u_{\zeta}(t,\zeta,y),\quad t\geq 0,\;\zeta\not=\xi(t,y),\;y\in\overline\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(t,\xi(t,y),y)=u_{*},\quad [\partial u/\partial\nu](t,\xi(t,y),y)=-1,\quad t\geq 0, y\in\overline\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\partial\xi/\partial\nu(t,y)=0,\quad t\geq 0,\;\zeta\not=\xi(t,y),\;y\in\partial\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(t,-\infty,y)=0,\quad u(t,\infty,y)=u_{\infty},\quad t\geq 0,\quad y\in\overline\Omega,NEWLINE\]NEWLINE was studied in [\textit{C.-M. Brauner, A. Lunardi} and \textit{Cl. Schmidt-Lainé}, Nonlinear Anal., Theory Methods Appl. 19, 455-474 (1992; Zbl 0780.35115), Appl. Math. Lett. 7, 1-4 (1994; Zbl 0814.35148), Nonlinear Anal., Theory Methods Appl. 44A, No. 2, 263-280 (2001; Zbl 1032.35082)]. NEWLINENEWLINENEWLINETravelling wave solutions are special solutions of the problem stated above such that \(\xi(t,y)=-c t +s(y)\), \(u(t,\zeta,y)=U(\zeta+c t,y)\). For certain values of parameters \(u_{\infty}, u_{*}\) this problem admits a unique (up to translations) one-dimensional or `planar' travelling wave solution \(c=c_0, s\equiv 0, U(z,y)=U_0(y)\). The purpose of this paper is to prove that in dimension \(n=2\), there exists a convergent (from the right) sequence \(u^k_{*}\to u_{*}^c\) of bifurcation points giving rise to branches of nonplanar travelling waves \((c,s,U)\) bifurcating from the `trivial branch' \((c_0,0,U_0)\). The fact that a sequence of bifurcation points accumulates at \(u^{c}_{*}\) contributes to understanding of the sharp instability phenomenon occuring at \(u_{*}=u^{c}_{*}\). NEWLINENEWLINENEWLINEThe proof of the main result is based on the Crandall-Rabinowitz theorem for bifurcation from simple eigenvalues [see \textit{M. G. Crandall} and \textit{P. H. Rabinowitz}, J. Funct. Anal. 8, 321-340 (1971; Zbl 0219.46015)]. However, its use is not straightforward since the dimension of the kernel and the co-dimension of the range of the linearized operator are two. To overcome this problem, the authors use some tricks, e.g., the translation invariance to reduce the dimension of the kernel.