Sharp pointwise estimates for the gradients of solutions to linear parabolic second-order equation in the layer

Abstract

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second-order equations with real constant coefficients in the layer ${\mathbb{R}}_T^{n+1}={\mathbb{R}}^n\times (0,T)$, where $n\ge 1$ and $T<\mathrm{\infty }$. The homogeneous equation is considered with initial data in $L^p\left({\mathbb{R}}^n\right)$, $1\le p\le \mathrm{\infty }$. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to $f\in L^p\left({\mathbb{R}}_T^{n+1}\right)\cap C^{\alpha }(\overline{{\mathbb{R}}_T^{n+1}})$, p>n + 2 and $\alpha \in (0,1)$. Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained.

COMMUNICATED BY:

• Robert Pertsch Gilbert

Keywords:

• Cauchy problem
• pointwise estimates for the gradient
• parabolic equation of the second order with constant coefficients

AMS Subject Classifications:

• Primary: 35K15
• Secondary: 35E99

Acknowledgements

The publication has been prepared with the support of the ‘RUDN University Program 5–100’.

Disclosure statement

No potential conflict of interest was reported by the authors.