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Estimates of the stabilization rate as $$t\to\infty$$ of solution of the first mixed problem for a quasilinear system of second-order parabolic equations. (English. Russian original) Zbl 0957.35068
Sb. Math. 191, No. 2, 235-273 (2000); translation from Mat. Sb. 191, No. 2, 91-131 (2000).
From the authors abstact and introduction: The authors study the stabilization of solutions of the first mixed problem in a cylindrical domain $$D=\{t>0\}\times\Omega$$ for the second-order quasilinear parabolic system $u_t= \text{div}(A(t,x,\triangledown u))-a(t,x,u),\quad (t,x)\in D,$ $u|_{\{t>0\}\times\partial\Omega}=0, \qquad u|_{t=0}=\varphi(x).$ Here $$u(t,x)$$ and $$\varphi(x)\in L_2(D)$$ are $$N$$-dimensional vector-valued functions, $$\Omega$$ is an arbitrary unbounded domain of $$\mathbb{R}^n$$, $$n\geqslant 2$$. The matrix $$A(t,x,\xi)$$, whose entries are measurable in $$(t,x)\in D$$, for all $$\xi,\eta\in \mathbb{R}^N$$ must satisfy the conditions $(A(t,x,\xi)-A(t,x,\eta)):(\xi-\eta)\geqslant c_1|\xi-\eta|^2,$ $|A(t,x,\xi)-A(t,x,\eta)\leqslant c_2|\xi-\eta|,\quad A(t,x,0)\equiv 0$ for almost all $$(t,x)\in D$$; $$\xi:\eta= \sum_{k=1}^N \sum_{i=1}^n \xi_{k,i}\cdot \eta_{k,i}$$. The components of the vector-valued function $$a(t,x,u)$$ are measurable in $$(t,x)\in D$$ and for all $$u,v\in \mathbb{R}^N$$ must satisfy the conditions $(a(t,x,u)-a(t,x,v))\cdot(u-v)\geqslant 0,$ $|a(t,x,u) -a(t,x,v)|\leqslant c_3|u-v|(|u |+|v|)^{q^*},\quad 0\leqslant q^*\leqslant\tfrac{4}{n}, \quad a(t,x,0)\equiv 0$ for almost $$(t,x)\in D$$, where $$u\cdot v=\sum_{k=1}^N u_kv_k$$, $$|u|^2=u\cdot u$$.
In a broad class of unbounded domains $$\Omega$$ two geometric characteristics of a domain are identified which determine the rate of convergence to zero as $$t\rightarrow\infty$$ of the $$L_2$$ norm of a solution. Estimates of the derivatives and uniform estimates of the solution are obtained; they are proved to be best possible in the order of convergence to zero in the case of one semilinear equation.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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