holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

are electroencephalogram eeg signals pseudo random number generators abstract in this paper we study if an electroencephalogram eeg signal can be treated as a pseudo random number generator prng we approach this problem by calculating the frequency of both healthy and epileptic eeg signals and we find that all eeg signals obey the gaussian distribution with different standard deviations however under some transformation we can treat eeg signals as prngs our transformation takes the sum of the least significant five bits of the eeg sample amplitude and then outputs the parity of this sum to a binary 0 or 1 sequence our binary sequences have passed nearly all nist pseudo random number tests except a few failures this indicates that the eeg signals can indeed become a prng provided that it undergoes a transformation like ours

intrinsic harnack inequalities for parabolic equations with variable exponents abstract in this paper we discuss a class of parabolic equations with variable exponents by using the steklov average and young s inequality we establish energy estimates for solutions to these equations then based on measure theoretical arguments we derive intrinsic harnack inequalities for solutions from functions and cylinders constructed here

harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

holder regularity and harnack inequality for degenerate parabolic equations related to caffarelli kohn nirenberg inequalities abstract this paper deals with the following parabolic equations 1 u t div x p u p 2 u f x t u in t 0 t where p 2 0 r n is a regular bounded domain containing the origin and f is a function with suitable hypothesis that we will precise later in order to obtain some weak harnack inequality

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains we study the positivity and regularity of solutions to the fractional porous me dium equations ut sum 0 in 0 for m 1 and s 0 1 with dirichlet boundary data u 0 in 0 n and nonnegative initial condition u 0 u0 0 our first result is a quantitative lower bound for solutions that holds for all positive times t 0 as a consequence we find a global harnack principle stating that for any t 0 solutions are comparable to ds m where d is the distance to this is in sharp contrast with the local case s 1 where the equation has finite speed of propagation after this we study the regularity of solutions we prove that solutions are classical in the interior c in x and c1 in t and establish a sharp cxs m regularity estimate up to the boundary our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form ut f u 0 in both in bounded and unbounded domains 2016 wiley periodicals inc

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

a harnack inequality for weighted degenerate parabolic equations abstract in this paper we generalize the recent result of dibenedetto gianazza vespri on the harnack inequality for degenerate parabolic equations to the case of a weighted p laplacian type operator in the spatial part the weight is assumed to belong to the suitable muckenhoupt class

harnack estimates for quasi linear degenerate parabolic differential equations we establish the intrinsic harnack inequality for non negative solutions of a class of degenerate quasilinear parabolic equations including equations of the p laplacian and porous medium type it is shown that the classical harnack estimate while failing for degenerate parabolic equations it continues to hold in a space time geometry intrinsic to the degeneracy the proof uses only measure theoretical arguments it reproduces the classical moser theory for non degenerate equations and it is novel even in that context holder estimates are derived as a consequence of the harnack inequality the results solve a long standing problem in the theory of degenerate parabolic equations

on time dependent domains for the degenerate p parabolic equation carleson estimate and holder continuity in this paper we propose a definition of parabolic nta for solutions to the degenerate p parabolic equation given this definition we prove the carleson estimate originally proved for this equation in avelin et al j eur math soc 2015 for cylindrical domains moreover we study a non optimal stronger outer corkscrew condition such that we obtain holder continuity up to the boundary for non negative solutions vanishing on a part of the boundary

subpotential lower bounds for nonnegative solutions to certain quasi linear degenerate parabolic equations nonnegative weak solutions of quasilinear degenerate parabolic equations of p laplacian type are shown to be locally bounded below by barenblatt type subpotentials as a consequence nonnegative solutions expand their positivity set that is a quantitative lower bound on a ball b at time t yields a quantitative lower bound on a ball b2 at some further time t these lower bounds also permit one to recast the harnack inequality of 1 in a family of alternative equivalent forms references 1 e dibenedetto u gianazza and v vespri harnack estimates for quasi linear degenerate parabolic differential equations acta math in press

harnack estimates for a quasi linear parabolic equation with a singular weight abstract in this paper based on measure theoretical arguments we establish harnack estimates and holder continuity of nonnegative weak solutions for a degenerate parabolic equation with a singular weight we transform the equation by performing the change of function the energy estimates the upper boundedness the lower boundedness and the expansion of positivity for the solutions to the transformed equation are obtained then our aim is reached

unbounded supersolutions of some quasilinear parabolic equations a dichotomy abstract we study unbounded supersolutions of the evolutionary p laplace equation with slow diffusion they are the same functions as the viscosity supersolutions a fascinating dichotomy prevails either they are locally summable to the power p 1 n p 0 or not summable to the power p 2 there is a void gap between these exponents those summable to the power p 2 induce a radon measure while those of the other kind do not we also sketch similar results for the porous medium equation

on the evolution governed by the infinity laplacian we investigate the basic properties of the degenerate and singular evolution equation ut d 2 u du du du du which is a parabolic version of the increasingly popular infinity laplace equa tion we prove existence and uniqueness results for both dirichlet and cauchy problems establish interior and boundary lipschitz estimates and a harnack inequality and also provide numerous explicit solutions

an asymptotic expansion for the expected number of real zeros of real random polynomials spanned by opuc abstract let i i 0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive borel measure that is symmetric with respect to conjugation we study asymptotic behavior of the expected number of real zeros say e n of random polynomials p n z i 0 n i i z where 0 n are i i d standard gaussian random variables when is the acrlength measure such polynomials are called kac polynomials and it was shown by wilkins that e n d admits an asymptotic expansion of the form e n d 2 log n 1 p 0 a p n 1 p kac himself obtained the leading term of this expansion in this work we generalize the result of wilkins to the case where is absolutely continuous with respect to arclength measure and its radon nikodym derivative extends to a holomorphic non vanishing function in some neighborhood of the unit circle in this case e n admits an analogous expansion with the coefficients a p depending on the measure for p 1 the leading order term and a 0 remain the same