We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 52, No. 3, pp. 47–58, July–September, 2009.
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Dmytryshyn, Y.B. A dynamic boundary-value problem without initial conditions for almost linear parabolic equations. J Math Sci 171, 474–489 (2010). https://doi.org/10.1007/s10958-010-0151-3
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DOI: https://doi.org/10.1007/s10958-010-0151-3