The unique shape of parabolic skis allows novices and intermediate skiers to master difficult turns more easily.
The first quantitative result of strong unique continuation for parabolic equations was derived in 1974 in the literature [1].
After 1988, there were more results of unique continuation for parabolic equations, and we refer the reader to [2 9], and the rich work cited therein.
We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space.
In [1], the authors establish the unique continuation for the parabolic equations with time independent coefficients in terms of the eigenfunctions of the corresponding elliptic operator, and their results did not apply to parabolic equations with time dependent coefficients.
Detection of distance-integrated, rather than localized, distributions of energy is a unique characteristic of these parabolic diffusion-wave fields.
Then we can find a unique u solving the following parabolic equation: left { textstylebegin{array}l@{quad}l} u_{t}=nablacdot (D tilde{u})nabla u )+nablacdot [ -chi nabla v+xinabla w)u ],& xinOmega, tin(0,T), frac{partial u}{partialnu}=0, & xinpartialOmega, tin (0,T), u(x,0)= u_{0}(x), &xinOmega.
The first is that the abstract result for the existence of the unique solution of certain nonlinear parabolic equation is obtained by using the properties of H-monotone operators, consequently, the proof is simplified compared to the corresponding discussions in the literature.
This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces.
