focusing blow up for quasilinear parabolic equations we study the behaviour of the non negative blowing up solutions to the quasilinear parabolic equation with a typical reaction diffusion right hand side and with a singularity in the space variable which takes the form where m 1 p 1 are arbitrary constants in the critical exponent case q p 1 m 0 we impose zero dirichlet boundary conditions at the singular point x 0 and at x 1 and take large initial data for a class of concave initial functions we prove focusing at the origin of the solutions as t approaches the blow up time t in the sense that x 0 belongs to the blow up set the proof is based on an application of the intersection comparison method with an explicit separable solution which has the same blow up time as u the method has a natural generalisation to the case of more general nonlinearities in the equation a description of different fine structures of blow up patterns in the semilinear case m 1 and in the quasilinear one m 1 is also presented a numerical study of the semilinear equation is also made using an adaptive collocation method this is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour

the blow up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source abstract the cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form u t d i v x u m 1 d u 1 d u u p is studied global in time existence and nonexistence conditions are found for a solution to the cauchy problem exact estimates of a solution are obtained in the case of global solvability a sharp universal i e independent of the initial function estimate of a solution near the blow up time is obtained

multi bump solutions for fractional nirenberg problem abstract we consider the multi bump solutions of the following fractional nirenberg problem 0 1 s u k x u n 2 s n 2 s u 0 in r n where s 0 1 and n 2 2 s if k is a periodic function in some k variables with 1 k n 2 s 2 we proved that 0 1 has multi bump solutions with bumps clustered on some lattice points in r k via lyapunov schmidt reduction it is also established that the 0 1 has an infinite many bump solutions with bumps clustered on some lattice points in r n which is isomorphic to z k

on a nirenberg type problem involving the square root of the laplacian abstract through an euler hopf type formula we establish existence result to a nirenberg type problem involving the square root of the laplacian in sphere s 2

a perturbation result for the qγ curvature problem on sn we consider the problem of prescribing the q curvature on sn using a perturbation method we obtain existence results for curvatures close to a positive constant

support properties of solutions to a degenerate equation with absorption and variable density abstract in this paper we study the properties of solutions to a degenerate parabolic equation with variable density and absorption we first obtain a critical exponent which distinguishes the localization of solutions from the positivity of them when positivity prevails we obtain the other critical exponent with respect to the decay of the variable density which separates the global existence of interfaces from the disappearance of them moreover the long time behavior of interfaces is characterized

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

blow up phenomena for polytropic equation with inhomogeneous density and source the subject of this investigation is the blow up phenomena of the positive solutions of the mixed problem for the one dimensional polytropic filtration equation with inhomogeneous density and source it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time note that the technique applied for the proof does not use the zel dovich kompaneets barenblatt solutions since the construction of such type of function is more complicated in our case therefore we obtain a result by multiplying on a special factor which has convenient properties in particular by choosing the parameters of the factor and using the properties of the solution we obtain the inequality which allows us to show the blow up phenomena

uniqueness of types of infinitely many bump solutions for the fractional nirenberg problem abstract in this study we fix k z 1 n 2 s 2 we consider the infinitely many bump solutions clustered near some lattice that is isomorphic to z k for the following fractional nirenberg problem s u k x u n 2 s n 2 s u 0 in r n we establish that if s 1 2 1 n 2 2 s then these types of solutions are unique and periodic in the first k variables

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

sharp short and long time l bounds for solutions to porous media equations with homogeneous neumann boundary conditions abstract we study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains r n n 3 with homogeneous neumann boundary conditions we improve the known results in such model case proving sharp l q 0 l regularizing properties of the evolution for short time and sharp long time l bounds for convergence of solutions to their mean value the generality of the discussion allows to consider almost at the same time weighted versions of the above equation provided an appropriate weighted sobolev inequality holds in fact we show that the validity of such weighted sobolev inequality is equivalent to the validity of a suitable l q 0 l bound for solutions to the associated weighted porous media equation the long time asymptotic analysis relies on the assumed weighted sobolev inequality only and allows to prove uniform convergence to the mean value with the rate predicted by linearization in such generality

peak solutions for the fractional nirenberg problem abstract we consider the fractional nirenberg problem u k u n 2 n 2 u 0 in r n assume k x 1 e k x and k x has at least two critical points satisfying some local conditions the existence of two peak solutions is obtained when the positive number e is small enough

existence results for the fractional nirenberg problem abstract we consider the fractional nirenberg problem on the standard sphere s n with n 4 using the theory of critical points at infinity we establish an euler hopf type formula and obtain some existence results for curvature satisfying assumptions of bahri coron type

smoothing effects for the filtration equation with different powers abstract we study the nonlinear diffusion equation u t u on general euclidean domains with homogeneous neumann boundary conditions we assume that u is bounded from below by u m 1 1 for small u and by u m 2 1 for large u the two exponents m 1 m 2 being possibly different and larger than one the equality case corresponds to the well known porous medium equation we establish sharp short and long time l q 0 l smoothing estimates similar issues have widely been investigated in the literature in the last few years but the neumann problem with different powers had not been addressed yet this work extends some previous results in many directions

asymptotic expansion of solutions to the cauchy problem for doubly degenerate parabolic equations with measurable coefficients abstract we study the asymptotic behaviour of nonnegative solutions of the cauchy problem for doubly degenerate parabolic equations with variable coefficients when the initial datum has a finite mass the asymptotic expansion of the solution for a large time uniformly in whole space is established

uniqueness and non uniqueness of bounded solutions to singular nonlinear parabolic equations we investigate well posedness of initial boundary value problems for a class of nonlinear parabolic equations with variable density at some part of the boundary called singular boundary the density can either vanish or diverge or not need to have a limit we provide simple conditions for uniqueness or non uniqueness of bounded solutions depending on the behaviour of the density near the singular boundary

well posedness of initial value problems for singular parabolic equations abstract we study well posedness of initial value problems for a class of singular quasilinear parabolic equations in one space dimension simple conditions for well posedness in the space of bounded nonnegative solutions are given which involve boundedness of solutions of some related linear stationary problems by a suitable change of unknown the above results can be applied to classical initial boundary value problems for parabolic equations with singular coefficients as the heat equation with inverse square potential

a new boundary element approach for contact problems with friction a boundary element solution algorithm for two dimensional and axisymmetric contact problems with friction based on an independent discretization of the contacting surfaces and under static and proportional loading conditions is presented the solution procedure uses the element shape functions to distribute the geometry tractions and displacements on each contact element the contact constraints are then applied between each contacting node and the opposite contact segment the overall boundary element matrix equations for the contacting bodies are coupled using the contact conditions at the interface