Potential theory and Schauder estimate in Hölder spaces for $(3 + 1)$-dimensional Benjamin–Bona–Mahoney–Burgers equation

The Cauchy problem for the well-known Benjamin–Bona–Mahoney–Burgers equation in the class of Hölder initial functions from $\mathbb{C}^{2+\alpha}(\mathbb{R}^3)$ with $\alpha\in(0,1]$ is considered. For such initial functions, it is proved that the Cauchy problem has a unique time-unextendable classical solution in the class $\mathbb{C}^{(1)}([0,T];\mathbb{C}^{2+\lambda}(\mathbb{R}^3))$ for any $T\in(0,T_0)$; moreover, either $T_0=+\infty$ or $T_0<+\infty$ 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.