1,720,986 research outputs found
Sur la non-unicite des solutions faibles de l'equation de la chaleur non lineaire avec non-linearite u^3
No full textWe prove for some singular initial data the existence of a solution u(t)=u(t)+v(t) ε C([0,T]:L3(ø3)) of the nonlinear heat equation with nonlinearity u3, which is not equal to Weissler's solution. The proof lies on the study of the perturbed equation on v(t) in weak-L6.Pour certaines données initiales singulières u0 ε L3(R3) nous prouvons l'existence d'une solution faible u(t)=u0+v(t.) ε C([0, T]; L3(R3)) de l'équation de la chaleur non linéaire avec non-linéarité u3 qui ne coïncide pas avec la solution de Weissler. La démonstration repose sur l'étude de l'équation perturbée sur v(t) dans L6-faible
Nonuniqueness for a critical nonlinear heat equation with any initial data.
No full textWe establish, for any initial data , with n3, the existence of an infinite number of solutions of the Cauchy problem for the nonlinear heat equatio
Non-radial maximizers for functionals with exponential non-linearity in R-2
No full textWe consider the functional F:H-0(1)(B(0,1))-> R
F(u)=integral(B(0,1)) vertical bar x vertical bar(alpha)(e(p vertical bar u vertical bar gamma)-1-p vertical bar u vertical bar(gamma))dx
where alpha>0, p>0, 1<= 2, and B(0,1) is the unit ball in R-2. We prove that for any p>0, 1<2 and 0<4 pi, gamma=2 no maximizer of F(u) on the unit ball in H-0(1) is radially symmetric provided that alpha is large enough. This extends a result of Smets, Su and Willem concerning the existence of non-radial ground state solutions for the Rayleigh quotient related to the Henon equation with Dirichlet boundary conditions
Non-uniqueness for a critical nonlinear heat equation
No full textHere we consider a class of non-linear heat equation with polynomial non-linearity. We prove a non-uniqueness result for mild solutions which take values in a critical Lebesgue space. To this end we extend to the entire space a counter-example of Ni and Sacks in the case where the underlying space is the ball of center 0 and of radius 1. We also propose a new criterion of uniqueness optimal with respect to the given counter-examples. The proof of our results lie on some estimates for the heat kernel in Lorentz spaces introduced by Meyer in the Navier–Stokes context
Heat equation with an exponential nonlinear boundary condition in the half space
Full text linkWe consider the initial-boundary value problem for the heat equation in the half space with an
exponential nonlinear boundary condition. We prove the existence of global-in-time solutions
under the smallness condition on the initial data in the Orlicz space expL2(RN
+ ). Furthermore,
we derive decay estimates and the asymptotic behavior for small global-in-time solutions
Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in R^2
No full textWe consider a semilinear heat equation with exponential nonlinearity in R2. We prove that local solutions do not exist for certain data in the Orlicz space exp L2(R2), even though a small data global existence result holds in the same space exp L2(R2). Moreover, some suitable subclass of exp L2(R2) for local existence and uniqueness is proposed
Besov spaces and unconditional well-posedness for the nonlinear Schrodinger equation in H-s (R-n)
No full textWe extend some results about uniqueness, without any extra condition, of solutions for the nonlinear Schrödinger equation with polynomial nonlinearities in low dimensions. The proof lies on paraproduct techniques and Besov spaces
Remarks on the H theorem for a non involutive Boltzmann like kinetic model
No full textIn this paper, we consider a one-dimensional kinetic equation of Boltzmann type in which the binary collision process is described by the linear transformation v* = pv + qw, w* = qv + pw, where (v, w) are the pre-collisional velocities and (v*, w*) the post-collisional ones and p ≥ q > 0 are two positive parameters. This kind of model has been extensively studied by Pareschi and Toscani (in J. Stat. Phys., 124(2–4):747–779, 2006) with respect to the asymptotic behavior of the solutions in a Fourier metric. In the conservative case p2 + q2 = 1, even if the transformation has Jacobian J ≠ 1 and so it is not involutive, we remark that the H Theorem holds true. As a consequence we prove exponential convergence in L1 of the solution to the stationary state, which is the Maxwellian
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