On the Cauchy problem for \(2\vec{b}\)-parabolic systems with increasing coefficients (Q2722246)

Let \(s\) be the l.c.m. of natural numbers \(b_1,\ldots,b_n\). Denote \(m_j:=s/b_j\), \({\mathbf k}:=(k_1,\ldots,k_n)\) \((k_j\in \mathbb Z_+)\), \(\|{\mathbf k}\|=\sum_{j=1}^{n}m_jk_j\).NEWLINENEWLINENEWLINEThe paper deals with the so-called \(2\vec{b}\)-parabolic system of the form NEWLINE\[NEWLINE\alpha(t)u_t-\beta(t)\sum_{\|{\mathbf k}\|\leq 2s}a_{{\mathbf k}}(t,x) \partial _{x}^{{\mathbf k}}u=0,\quad(t,x)\in (0,T]\times \mathbb R^n,\tag{1}NEWLINE\]NEWLINE where \(u\in \mathbb C^n\), \(\alpha(t),\beta(t)\in C((0,T], [0,\infty))\), \(\alpha(0)\beta(0)=0\), and \(a_{{\mathbf k}}(t,x)\) are \(N\times N\)-matrices. It is assumed that the system (1) with \(\alpha(t)\equiv\beta(t)\equiv 1\) is dissipative with the characteristic of dissipation \(D(x)\). Such dissipative parabolic systems with degeneration on the initial hyperplane \(t=0\) were studied earlier by the authors in [Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 1999, No. 6, 18-22 (1999; Zbl 0936.35077)].NEWLINENEWLINENEWLINENow the case of increasing coefficients (as \(\|x\|\to \infty \)) is analyzed in detail. The authors prove the existence of the fundamental matrix \(Z(t,x;\tau,\xi)\) to (1). Such a matrix allows to represent the solution \(u(t,x)\), which satisfies the initial condition \(u|_{t=\tau }=\varphi(x)\), in the form NEWLINE\[NEWLINEu(t,x)=\int_{\mathbb R^n}Z(t,x;\tau,\xi)\varphi(\xi) d\xi.NEWLINE\]NEWLINENEWLINENEWLINENEWLINEBy the using fundamental matrix the representations and estimates for solutions of non-homogeneous Cauchy problems are also obtained.