Gradient estimates of the solution to the Cauchy problem for degenerate parabolic equations with nonpower nonlinearities

ABSTRACT

We prove the uniform gradient estimates of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The large-time estimates in terms of initial data mass and nonlinearity structure were obtained. In the particular case, when the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound.

KEYWORDS:

• Cauchy problem
• degenerate parabolic equations
• nonpower nonlinearity
• gradient estimates
• long-time behavior

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35B45
• Secondary: 35B45
• 35K55
• 35K59
• 35K65

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by North-Caucasus Center for Mathematical Research of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, agreement No 075-02-2023-914.