Summary: The following initial-boundary value problem is considered \[ \begin{aligned} & E(t,x,u)u_ t+\sum^ n_{i=1} A_ i(t,x,u)u_{x_ i}+B(t,x,u)=F(t,x,u)\quad\text{in }\Omega^ T,\\ & M(t,x,u)u|_{\partial\Omega}=g(t,x,u)\quad\text{on }\partial\Omega^ T,\quad u|_{t=0}=u_ 0\quad\text{in }\Omega,\end{aligned}\tag{1} \] where \(E,A_ 1,\dots,A_ n\) are symmetric matrices, \(m\times m\), \(E\), \(B\) are positive definite, \(u\in G\subset\mathbb{R}^ m\), \(x\in\Omega\subset\mathbb{R}^ n\). The range of \(M\) is a subspace of \(\mathbb{R}^ m\) generated by eigenvectors of the matrix \(- A_{\bar n}=-(A_ 1 n_ 1+\cdots+A_ n n_ n)\), where \(\bar n\) is the unit outward vector normal to \(\partial\Omega\), which correspond to positive eigenvalues. Having found global in time a priori estimate for sufficiently small data in Sobolev spaces: \[ {\mathcal H}^ s\equiv\bigcap^ s_{i=1} L^ i_ \infty(0,T;H^{s-i}(\Omega))\cap H^ s(\Omega^ T)\cap H^ s(\partial\Omega^ T),\quad T\in\mathbb{R}_ +,\;s>n/2+1, \] for coefficients from \({\mathcal H}^ s\) too, the existence of the linearized problem (1) is found in these spaces and then by the method of successive approximations the existence of global solutions to (1) is shown in the same spaces, too.