On the global solvability of the Cauchy problem for a quasilinear ultraparabolic equation (Q2852326)

The solvability of the following ultraparabolic equation NEWLINE\[NEWLINE{u_t} + g(t,u){u_x} + f(t,u) = {\Delta _y}u\quad \text{in }\, S_{T}=(0,T)\times\mathbb R \times\mathbb R^n, \tag{1}NEWLINE\]NEWLINEcoupled with initial condition NEWLINE\[NEWLINEu(0,x,y) = u_0(x,y), \tag{2}NEWLINE\]NEWLINEwhere \( {\Delta _y}=\sum\nolimits_1^n {{{{\partial ^2}} / {\partial y_i^2}}} \), is considered. Equation (1) describes nonstationary transport (of matter, impulse, temperature) processes where in some direction the effect of the diffusion is negligible as compared to the convection. Such equations appear in age dependent population diffusion and in mathematical finance. This class of equations has received considerable attention in the recent decades from different authors.NEWLINENEWLINEThe local existence of a smooth solution (at least Lipschitz continuous with respect to \(x,\,y\) and Hölder continuous with respect to \(t)\) to problem (1), (2) was proved by Pascucci and Polidoro under the assumptions that \(g,\) \(f\) and \({u_0}\) are globally Lipschitz continuous functions, the global classical solvability was obtained if additionally \({u_0}\) is nonincreasing with respect to \(x\) and \(g(u) - g(v) \geqslant {c_0}(u - v)\) for some positive constant \({c_0}.\) Main goal of this paper is to prove the global solvability of considered problem for locally Hölder continuous functions \(g\) and \(f\) under the following structure restriction: NEWLINE\[NEWLINEK\left| {g(t,{u_2}) - g(t,{u_1})} \right| \leqslant f(t,{u_2}) - f(t,{u_1})\quad \text{for } \,{u_1} < {u_2},\,\,\,t \in [0,T), \tag{3}NEWLINE\]NEWLINEwhere \(T > 0\) is an arbitrary positive constant and NEWLINE\[NEWLINEk = \sup_{R \times {R^n}} \frac{{{u_0}(x,y) - {u_0}(x',y)}}{{\left| {x - x'} \right|}}. NEWLINE\]NEWLINENEWLINEIt is showed that the presence of low order term \(f\) connected with \(g\) as in (3) provides global solvability, and it in some sense is optimal.