The initial value problem for parabolic equations with data in
Eugene Fabes (1972)
Studia Mathematica
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Eugene Fabes (1972)
Studia Mathematica
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Philippe Souplet, Slim Tayachi (2001)
Colloquium Mathematicae
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Consider the nonlinear heat equation (E): . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality . More general inequalities of the form with, for instance, are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...
M. Guedda (2002)
Colloquium Mathematicae
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We study the absence of nonnegative global solutions to parabolic inequalities of the type , where , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....
Ahmed Aberqi, Jaouad Bennouna, M. Hammoumi, Mounir Mekkour, Ahmed Youssfi (2014)
Applicationes Mathematicae
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We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ in Ω. with s = (N+2)/(N+p) (p-1), , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), and f ∈ L¹(Q).
Grzegorz Karch (1997)
Annales Polonici Mathematici
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We study the decay in time of the spatial -norm (1 ≤ p ≤ ∞) of solutions to parabolic conservation laws with dispersive and dissipative terms added uₜ - uₓₓₜ - νuₓₓ + buₓ = f(u)ₓ or uₜ + uₓₓₓ - νuₓₓ + buₓ = f(u)ₓ, and we show that under general assumptions about the nonlinearity, solutions of the nonlinear equations have the same long time behavior as their linearizations at the zero solution.
P. Besala (1971)
Annales Polonici Mathematici
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Abderrahmane El Hachimi, Jaouad Igbida, Ahmed Jamea (2010)
Applicationes Mathematicae
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We study the existence of solutions of the nonlinear parabolic problem in ]0,T[ × Ω, on ]0,T[ × ∂Ω, u(0,·) = u₀ in Ω, with initial data in L¹. We use a time discretization of the continuous problem by the Euler forward scheme.
Andrzej Raczyński (2007)
Studia Mathematica
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The existence of solutions to a nonlinear parabolic equation describing the temporal evolution of a cloud of self-gravitating particles with a given external potential is studied in weak- spaces (i.e. Markiewicz spaces). The main goal is to prove the existence of global solutions and to study their large time behaviour.
J. W. Cholewa, T. Dłotko (2003)
Banach Center Publications
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An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on . This semigroup possesses an -global attractor that is closed, bounded, invariant in , and attracts bounded subsets of in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...
Ph. Souplet (2009)
Journal of the European Mathematical Society
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We consider positive solutions of the system ; in a ball or in the whole space, with . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...
Ludwik Byszewski (1990)
Annales Polonici Mathematici
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Cung The Anh, Le Thi Thuy (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...
Paweł Goldstein (2008)
Colloquium Mathematicae
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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be , and in the variable exponent case, L² and -weak.
Daniel Wachsmuth (2016)
Commentationes Mathematicae Universitatis Carolinae
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Let with be given. Then we show by means of a counter-example that the positive part of has less regularity, in particular it holds in general. Nevertheless, satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.
Anne-Laure Dalibard (2011)
Journal of the European Mathematical Society
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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type , where and the flux is periodic in . More specifically, we consider the case when the initial data is an disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in norm like a self-similar profile for large times. The proof uses a time and space change of variables...
Alessandra Lunardi (1998)
Studia Mathematica
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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.
Victor Galaktionov (2004)
Journal of the European Mathematical Society
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We consider th-order semilinear parabolic equations in , with Dirac’s mass as the initial function. We show that for , the Cauchy problem admits a solution which is bounded and smooth for small , while for such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.
Arkadiusz Szymaniec (2010)
Applicationes Mathematicae
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We consider the initial-value problem for a linear hyperbolic parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove time decay estimates for the solution of the associated linear Cauchy problem.
Šolín, Pavel, Segeth, Karel, Doležel, Ivo
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We present a new class of self-adaptive higher-order finite element methods (-FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methods do not contain any tuning parameters and work reliably with both low- and high-order finite elements. The methodology was used to solve various types of problems including thermoelasticity,...
Emmanuel P. Smyrnelis (2010)
Annales Polonici Mathematici
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The study of the equation (L₂L₁)*h = 0 or of the equivalent system L*₂h₂ = -h₁, L*₁h₁ = 0, where is a second order elliptic differential operator, leads us to the following general framework: Starting from a biharmonic space, for example the space of solutions (u₁,u₂) of the system L₁u₁ = -u₂, L₂u₂ = 0, being elliptic or parabolic, and by means of its Green pairs, we construct the associated adjoint biharmonic space which is in duality with the initial one.
Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)
Journal of the European Mathematical Society
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A regular normal parabolic geometry of type on a manifold gives rise to sequences of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative on the corresponding tractor bundle , where is the normal Cartan connection. The first operator in the sequence is overdetermined and it is well known that yields the prolongation of this operator in the homogeneous case . Our first...
Ludwik Byszewski (1990)
Annales Polonici Mathematici
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Joanna Rencławowicz (2003)
Banach Center Publications
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In this paper we examine self-similar solutions to the system , i = 1,…,m, , t > 0, , i = 1,…,m, , where m > 1 and , to describe asymptotics near the blow up point.