An estimate for the rate of convergence in the inverse Cauchy problem with rapidly oscillating periodic coefficients (Q2901642)

In the present paper the inverse Cauchy problem for the equation NEWLINE\[NEWLINE \frac{\partial u^{\varepsilon}(t,x)}{\partial t}+\frac{1}{2}\beta^2\biggl(\frac{x}{\varepsilon}\biggl)\frac{\partial^2 u^{\varepsilon}(t,x)}{\partial x^2}+\alpha\biggl(\frac{x}{\varepsilon}\biggl)\frac{\partial u^{\varepsilon}(t,x)}{\partial x}=f(t,x)NEWLINE\]NEWLINE with \(u^{\varepsilon}(T,x)=\varphi(x)\), \(0\leq t \leq T\), is considered. The coefficients are assumed to satisfy the following conditions: \(\alpha(x)=\alpha(x+1)\), \(\beta(x)=\beta(x+1)\), \(| \alpha(x)| +| \beta(x)| +| f(t,x)| +| \varphi(x)| \leq K <+\infty,\) \(0<\lambda \leq \beta^2(x)\leq K <+\infty\), \(| \alpha(x)-\alpha(y)| +| \beta(x)-\beta(y)| +| \varphi(x)-\varphi(y)| +| f(t,x)-f(t,y)| \leq L| x-y| \). Using methods from probability theory, the estimate \(| u^{\varepsilon}(t,x)-u(t,x)| \leq \sqrt{\varepsilon}C(K,L,T,\lambda)\) is obtained. Here \(u(t,x)\) denote the solution of the limiting problem.