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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.

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