Sharp pointwise estimates for solutions of weakly coupled second-order parabolic system in a layer

ABSTRACT

We deal with m-component vector-valued solutions to the Cauchy problem for a linear both homogeneous and nonhomogeneous weakly coupled second-order parabolic system in the layer ${\mathbb{R}}_T^{n+1}={\mathbb{R}}^n\times (0,T)$. We assume that coefficients of the system are real and depending only on t, $n\ge 1$ and $T<\mathrm{\infty }$. The homogeneous system is considered with initial data in $\left[L^p\right({\mathbb{R}}^n){]}^m$, $1\le p\le \mathrm{\infty }$. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to $\left[L^p\right({\mathbb{R}}_T^{n+1}){]}^m\cap [C^{\alpha }(\overline{{\mathbb{R}}_T^{n+1}}){]}^m$, $\alpha \in (0,1)$. Explicit formulas for the sharp coefficients in pointwise estimates for solutions to these problems and their directional derivative are obtained.

Keywords:

• Cauchy problem
• weakly coupled parabolic system
• sharp pointwise estimates
• directional derivative of a vector-valued function

AMS Subject Classifications:

• Primary 35K45
• 35A23
• Secondary 47A30

Acknowledgments

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 author(s).