We consider the Cauchy problem for the isotropic superdiffusion equation with the Riesz fractional differentiation operator of order $\alpha\in(0;2)$, which generalizes the classical heat conduction equation. Such models arise in the description of anomalous transport of energy and mass in fractal and porous media, plasma, and other complex structures exhibiting superdiffusive behavior. The Riesz operator is the generator of symmetric $\alpha$-stable Levy processes; therefore, the solution of the Cauchy problem may be interpreted as the probability density of the corresponding stochastic process.

We prove the existence of a classical bounded smooth solution, even when the initial data contain a finite number of integrable discontinuities of the second kind.

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