Estimate of rate of convergence of some quasilinear parabolic equations with weakly convergent coefficients (Q1123314)

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.