Parabolic equations with changing evolution direction (Q2760728)

In the domain \(Q=(-\infty,\infty)\times(0,T)\) the author considers the parabolic equation with changing evolution direction NEWLINE\[NEWLINE \text{sign} x\cdot u_t=(-1)^{n+1} {\partial^{2n}u\over\partial x^{2n}} NEWLINE\]NEWLINE (here \(n\) is a nonnegative integer). He solves the problem of finding a solution to this equation. The author seeks this solution in the Hölder space \(H^{p,p/2n}_{xt}\), with \(p=2nl+\gamma\), \(0<\gamma<1\), \(l\geq 0\) an integer, under the initial conditions NEWLINE\[NEWLINEu(x,0)=\varphi_1(x),\quad x>0, \qquad u(x,T)=\varphi_2(x),\quad x<0,NEWLINE\]NEWLINE and compatibility conditions NEWLINE\[NEWLINE {\partial^k u\over\partial x^k} (-0,t)= {\partial^k u\over\partial x^k} (+0,t),\quad k=0,1,\dots,2n-1. NEWLINE\]NEWLINE The author writes out the orthogonality conditions under which the problem has a unique solution.