### On the Classical Solvability of Mixed Problems for a Second–Order One–Dimensional Parabolic Equation

#### Abstract

{\bf Abstract.}\ \ We prove the existence and uniqueness of classical

solutions to mixed problems for the equation

$$

\frac{\partial u}{\partial t}(x,t)\ -\ \frac{\partial^2u}{\partial x^2}(x,t)\

+\ q(x)\,u(x,t)\ =\ f(x,t)

$$

on a rectangle $\ \overline\Omega\,=\,[a,b]\times [0,T]\,$,\ with arbitrary self--adjoint homogenous boundary conditions.\ We assume that $\ q$ \ and $\ f$ \ are continuous functions, that $\ f(x,\cdot)$ \ satisfies a H\"older condition uniformly with respect to $\ x\,$,\ and the initial function belongs to the class $\ \overset{\circ}\to W{}^{(1)}_p(a,b)\ (\,1< p\le 2\,)\,$.\ Also, an upper--bound estimate for the solution and, as a consequence, a kind of stabi\-lity of the solution with respect to the initial function are established.\ Moreover, some convergence rate estimates for the series defining solutions (\,and their first derivatives\,) are given.\ A modification of the Fourier method is used.

\smallpagebreak

Based on the obtained results, we also study the mixed problems on an unbounded rectangle $\ \overline{\Omega_{\infty}}\,=\,[a,b]\times [0,+\infty)\,$.\ The existence and uniqueness of classical solutions are established, and some properties of the solutions are considered.

solutions to mixed problems for the equation

$$

\frac{\partial u}{\partial t}(x,t)\ -\ \frac{\partial^2u}{\partial x^2}(x,t)\

+\ q(x)\,u(x,t)\ =\ f(x,t)

$$

on a rectangle $\ \overline\Omega\,=\,[a,b]\times [0,T]\,$,\ with arbitrary self--adjoint homogenous boundary conditions.\ We assume that $\ q$ \ and $\ f$ \ are continuous functions, that $\ f(x,\cdot)$ \ satisfies a H\"older condition uniformly with respect to $\ x\,$,\ and the initial function belongs to the class $\ \overset{\circ}\to W{}^{(1)}_p(a,b)\ (\,1< p\le 2\,)\,$.\ Also, an upper--bound estimate for the solution and, as a consequence, a kind of stabi\-lity of the solution with respect to the initial function are established.\ Moreover, some convergence rate estimates for the series defining solutions (\,and their first derivatives\,) are given.\ A modification of the Fourier method is used.

\smallpagebreak

Based on the obtained results, we also study the mixed problems on an unbounded rectangle $\ \overline{\Omega_{\infty}}\,=\,[a,b]\times [0,+\infty)\,$.\ The existence and uniqueness of classical solutions are established, and some properties of the solutions are considered.

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