Phragmén-Lindelöfs type theorems for the solutions of degenerate second-order parabolic equations of nonstationary diffusion-convections type with homogeneous Cauchy conditions (Q2720923)

Phragmen-Lindelöfs type theorems are proved for the generalized solutions of following Cauchy problem: NEWLINE\[NEWLINE \frac{\partial}{\partial t}(|u|^{q-2}u) -\sum_{i=1}^n\frac{d}{dx_i}a_i(t,x,u,\nabla_xu)+ \bigl(\overline a,\nabla_x b(t,u)\bigr)=0, \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0)=0,\tag{2}NEWLINE\]NEWLINE where \(q>1, \overline a\in \mathbb R^n\), \(n\geq 1\), \((t,x)\in (0,T) \times \mathbb R^n\), and Carathéodory's functions \(a_i(t,x,\eta,\xi)\) and the continuous function \(b(t,\eta)\) satisfy the following conditions: NEWLINE\[NEWLINE \sum_{i=1}^na_i(t,x,\eta,\xi)\xi_i \geq d_0|\xi|^{p}, \quad p>q, d_0>0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE |a_i(t,x,\eta,\xi)|\leq d_1|\xi|^{p-1}, \quad d_1<\infty, NEWLINE\]NEWLINE NEWLINE\[NEWLINE |b(t,\eta)|\leq d_2|\eta|^{\lambda-1}, \quad d_2<\infty , \lambda >q, NEWLINE\]NEWLINE NEWLINE\[NEWLINE |\eta|^{s}b(t,\eta)\eta-(s+\lambda)\int\limits_0^\eta b(t,\theta)|\theta|^s d\theta\geq d_3|\eta|^{\lambda +s}, \quad d_3>0, 0\leq s<\infty. NEWLINE\]NEWLINE There are found in some sense sharp low estimates of the speed of growth of nonzero generalized solutions for problem (1)--(2). This growth can be isotropic or unisotropic in dependence of conditions on parameters \(q\), \(p\), \(\lambda\).