Abstract
We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method of E.E. Levi.
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Acknowledgement
The first author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the project “sostegno alla Ricerca individuale”.
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Formica, M.R., Ostrovsky, E. & Sirota, L. Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coefficient at the Origin. Extension of the Classical Parametrix Method. Acta Appl Math 170, 399–413 (2020). https://doi.org/10.1007/s10440-020-00339-5
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DOI: https://doi.org/10.1007/s10440-020-00339-5
Keywords
- Partial Differential Equation of parabolic type
- Fundamental solution
- Generalized Mittag-Leffler function
- Chapman-Kolmogorov equation
- Neumann series
- Volterra’s integral equation