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Estimate of rate of convergence of some quasilinear parabolic equations with weakly convergent coefficients. (English. Russian original) Zbl 0677.35057

Sib. Math. J. 29, No. 5, 782-791 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 118-130 (1988).
The author gives an estimate of the convergence rate for the solutions of the following \(problems:\)
(1\({}^ m) \) \(d^ m(t,x,u)u_ t=a^ m(t,x)u_{xx}+b^ m(t,x,u)u_ x+q^ m(t,x)u_ x+p^ m(t,x,u)+f^ m(t,x)\), \(m=1,2,...,\infty,\)
(2) \(u(0,x)=u(t,0)=u(t,\ell)=0\), in the domain \[ Q=\{(t,x):\quad 0\leq t\leq T,\quad 0\leq x\leq \ell \},\quad and \]
(3\({}^ m) \) \(u_ t=a^ m(t,x)u_{xx}+b^ m(t,x)u_ x+p^ m(t,x)u+f^ m(t,x)\), \(m=1,2,...,\infty,\)
(4) \(u(0,x)=u_ x(t,0)=u(t,\ell)=0\), in the same domain Q.
Concerning the functions of these equations, it is supposed that: \(d^ m\), \(a^ m\), \(b^ m\) are measurable and founded on every compact, uniformly with respect to m, \(d^ m(t,x,u)\geq d_ 0=const>0,\) \(a^ m(t,x)\geq a_ 0=const>0,\) \(\forall (t,x,u)\in Q\times R\), \(m=1,2,...,\infty\), \(| a^ m(t,x)| \leq M_ 1\), \(\forall (t,x)\in Q\), \(m=1,2,...\), \(d^ m\to d^{\infty}\), \(q^ m\to q^{\infty}\), \(p^ m\to p^{\infty}\), \(f^ m\to f^{\infty}\) weak in \(L^ 2\) and \(a^ m\to a^{\infty}\), \(b^ m\to b^{\infty}\) uniformly.
Reviewer: I.Onciulescu

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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