Cauchy problem for the heat equation with Laplacian with respect to a measure

Dmytro M. Yakymets

Abstract


The purpose of this work is to build the heat equation in a space with finite variant measure, using the concept of divergence with respect to a measure, to get a solution of the initial vaule problem for some partial cases as well as the general fundamental solution, to prove uniqueness and existence of the solution of the corresponding initial value problem, to analyse the conditions the solution of the initial value problem exists and is unique under. The solutions for two partial cases are obtained by means of Fourier transformation, whereas general solution is obtained via the parametrix technique. The measure $\mu$ that is used in this work is considered as positive absolutely continuous w.r.t invariant Lebesgue measure, and its Radon–Nikodym derivative is piecewise smooth and is bounded with its first derivative on $\mathbb{R}^n$. If a vector field $\vb{Z}$ is smooth on $\mathbb{R}^n$, then $\mu$ is differentiable along $\vb{Z}$, and its logarithmic derivative along $\vb{Z}$ is denoted as $\dive_\mu \vb{Z}$. In this case Laplace operator is introduced on $C^2_b(\mathbb{R}^n)$ as $\Delta u = \dive_\mu ( \grnvect u$), and thereby we can set standard Cauchy problem for heat equation:

$ begin{cases}\dfrac{\partial u}{\partial{t}}=\Delta u; u(x,0)=\varphi(x), \end{cases},$

which is to be solved by means of aforementioned methods.

Keywords


Heat equation; Laplace operator; Variant measure; Initial value problem; Divergence with respect to a measure

References


Bogachev, V. I. (2010). Differentiable measures and the Malliavin calculus (No. 164). Providence, RI: American Mathematical Society.

Bogdanskii, Y. V., & Sanzharevskii, Y. Y. (2014). The Dirichlet problem with Laplacian with respect to a measure in the Hilbert space. Ukrainian Mathematical Journal, 66 (6), 818–826. https://doi.org/10.1007/s11253-014-0976-x

Belopolskaya, Y. I., & Daletsky, Y. L. (1990). Stochastic equations and differential geometry. Dordrecht: Kluwer Academic Publishers.

Friedman, A. (2008). Partial differential equations of parabolic type. Mineola, NY: Dover Publications.






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