On solvability of initial-boundary value problems for a nonlinear equation of variable type
The author studies solvability of the following boundary value problems: find a solution to the equation \[ uu_t-u_{xx}=f(x,t)\quad (0<x<l,\;0<t<T) \tag{1} \] satisfying the conditions \[ u|_{t=0}=u_0(x),\quad u|_{x=0}=\varphi_1(t),\quad u|_{x=l}=\varphi_2(t) \tag{2} \] or \[ u|_{t=0}=u_0(x),\quad u_x|_{x=0}=u_x|_{x=l}=0. \tag{3} \] Note that (1) is a nonlinear equation with variable time direction. The author proves that the boundary value problem (1), (2) has a unique regular solution provided that, in addition to some smoothness and agreement conditions, the following conditions hold: (i) \(u_0(x) \geq \delta > 0\), \(x\in[0,l]\), \(\varphi_i(t)\geq\delta>0\), \(t\in [0,T]\); (ii) \(l\) is small; or, alternatively, (i) and the condition (iii) the functions \(f(x,t)\), \(u_0(x)\), and \(\varphi_i(t)\) are small. The boundary value problem (1), (3) has a unique regular solution provided that, in addition to some smoothness and agreement conditions, the following conditions hold: (i\('\)) \(u_0(x)\geq\delta>0\), \(x\in[0,l]\); (iii\('\)) the functions \(f(x,t)\) and \(u_0(x)\) are small. The above smallness conditions on \(l\) or \(f(x,t)\), \(u_0(x)\), \(\varphi_1(t)\), \(\varphi_2(t)\) are found. Note that, unlike many preceding papers, the condition \(f(x,t)\geq 0\), guaranteeing that the solution preserves sign, is omitted.
scientific article; zbMATH DE number 1285462