Geodesic
Parabolic equations over the field of $p$-adic numbers
Izvestiya. Mathematics , Tome 39 (1992) no. 3, pp. 1263-1280

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The author constructs and investigates a fundamental solution of Cauchy's problem for a parabolic equation with a $p$-adic space variable and a real time variable. The question of existence and uniqueness of solutions to Cauchy's problem in classes of bounded and increasing functions is considered, and conditions for nonnegativity of the fundamental solution are found. The problem of determining if the solution stabilizes as $t\to\infty$ is solved for a model equation with constant coefficients. 
@article{IM2_1992_39_3_a8,
     author = {A. N. Kochubei},
     title = {Parabolic equations over the field of $p$-adic numbers},
     journal = {Izvestiya. Mathematics },
     pages = {1263--1280},
     publisher = {mathdoc},
     volume = {39},
     number = {3},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1992_39_3_a8/}
}
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A. N. Kochubei. Parabolic equations over the field of $p$-adic numbers. Izvestiya. Mathematics , Tome 39 (1992) no. 3, pp. 1263-1280. http://geodesic.mathdoc.fr/item/IM2_1992_39_3_a8/