Potential Theory and Schauder Estimate in Hölder Spaces for (3 + 1)-Dimensional Benjamin–Bona–Mahoney–Burgers Equation

Abstract: The Cauchy problem for the well-known Benjamin–Bona–Mahoney–Burgers equation in the class of Hölder initial functions from C2+α(R3) α (0,1] is considered. For such initial functions, it is proved that the Cauchy problem has a unique time-unextendable classical solution in the classC(1)([0,T];C2+λ(R3)) for any T (0T0) moreover, either T0 = +∞ T0 < +∞ and, in the latter case, T(0) is the solution blow-up time. To prove the solvability of the Cauchy problem, we examine volume and surface potentials associated with the Cauchy problem in Hölder spaces. Finally, a Schauder estimate is obtained. © 2021, Pleiades Publishing, Ltd.