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Riesz Potential Estimates for Parabolic Equations with Measurable Nonlinearities

https://doi.org/10.1093/imrn/rnx080

We study a nonlinear parabolic equation with measurable nonlinearity and measure data:  


u t − d i v a ( D u , u , x , t ) = μ i n Ω T .

We obtain an optimal point-wise estimate for the gradient of a solution in terms of a certain Riesz potential of the right-hand measure in a very general setting that the nonlinearity is not only allowed to be measurable in one of spatial variables, but also depends on the variable of the possibly unbounded solution itself.

We study a nonlinear parabolic equation with measurable nonlinearity and measure data:  


u t − d i v a ( D u , u , x , t ) = μ i n Ω T .

We obtain an optimal point-wise estimate for the gradient of a solution in terms of a certain Riesz potential of the right-hand measure in a very general setting that the nonlinearity is not only allowed to be measurable in one of spatial variables, but also depends on the variable of the possibly unbounded solution itself.