On a multidimensional strongly nonlinear equation of third order in non-cylindrical domains (Q2713901)

The author studies the solvability of boundary value problems for the third-order equations NEWLINE\[NEWLINE Lu=u_{tt} - \sum_{i=1}^n \frac{\partial}{\partial x_i}\varphi(\nabla u) - \Delta u_t = f(x,t).\tag{1} NEWLINE\]NEWLINE Let \(D_0\) be a bounded domain in \({\mathbb R}^n\) with smooth boundary. Put the domains \(D_{t}=\{x\in {\mathbb R}^n: (\frac{x_1}{\alpha_1(t)},\dots, \frac{x_1}{\alpha_n(t)})\in D_0\}\), \(Q_T=\{(x,t)\in {\mathbb R}^n: x\in D_t,\;0< t< T\}\), and \(S=\cup_{0<t<T}[\partial D_t\times \{t\}].\) Equation (1) is furnished with the boundary conditions NEWLINE\[NEWLINE u(x,0)=u_0(x),\;u_t(x,0)=u_1(x) (x\in D_0),\;\;u_t|_S=0 \tag{2} NEWLINE\]NEWLINE whenever \(\alpha_i'\leq 0\) for every \(i\) and with the conditions NEWLINE\[NEWLINE u(x,0)=u_0(x),\;u_t(x,0)=u_1(x) (x\in D_0),\;\;u_t|_S=u|_{S}=0 \tag{3} NEWLINE\]NEWLINE whenever \(\alpha_i(t)\equiv \alpha(t)\) for every \(i\) and \( \alpha'(t)>0\). Under some natural conditions on the nonlinearity \(\varphi(\nabla u)\) (smoothness and boundedness of the first derivatives), the conditions for the domains \(D_t\) (every domain \(D_t\) is close to a star-shaped domain in some sense), and smoothness conditions for the data of the problems, it is shown that the problem (1), (2) ((1), (3)) is solvable and a solution belongs to some Sobolev class; in particular, its generalized derivatives occurring in (1) belong to the space \(L_2(Q_T)\). The equality (1) is fulfilled almost everywhere on \(Q_T\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].