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1
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Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

100%
Karch G.
Studia Mathematica
|
2000
|
tom 143
|
nr 2
175-197
EN
Large time behavior of solutions to the generalized damped wave equation $u_{tt} +A u_t +ν B u+F(x,t,u,u_t,∇ u) = 0$ for $(x,t)∈ ℝ^n × [0,∞)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $Au_t = u_t$, $Bu = -Δu$, and the nonlinear term is either $|u_t|^{q-1}u_t$ or $|u|^{α-1}u$. In this case, the asymptotic profile of solutions is given by a multiple of the Gauss-Weierstrass kernel. Our method of proof does not require the smallness assumption on the initial conditions.
2
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Long-time asymptotics of solutions to some nonlinear wave equations

100%
Karch G.
Banach Center Publications
|
2000
|
tom 52
|
nr 1
133-146
EN
In this paper, we survey some recent results on the asymptotic behavior, as time tends to infinity, of solutions to the Cauchy problems for the generalized Korteweg-de Vries-Burgers equation and the generalized Benjamin-Bona-Mahony-Burgers equation. The main results give higher-order terms of the asymptotic expansion of solutions.
3
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$L^p$-decay of solutions to dissipative-dispersive perturbations of conservation laws

100%
Karch G.
Annales Polonici Mathematici
|
1997
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tom 67
|
nr 1
65-86
EN
We study the decay in time of the spatial $L^p$-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.
4
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Large time behaviour of solutions to nonhomogeneous diffusion equations

64%
Dolbeault J., Karch G.
Banach Center Publications
|
2006
|
tom 74
|
nr 1
133-147
EN
This note is devoted to the study of the long time behaviour of solutions to the heat and the porous medium equations in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csiszár-Kullback type estimates.
5
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Blow-up versus global existence of solutions to aggregation equations

64%
Karch G., Suzuki K.
Applicationes Mathematicae
|
2011
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tom 38
|
nr 3
243-258
EN
A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.
6
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A Neumann problem for a convection-diffusion equation on the half-line

64%
Biler P., Karch G.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
79-95
EN
We study solutions to a nonlinear parabolic convection-diffusion equation on the half-line with the Neumann condition at x=0. The analysis is based on the properties of self-similar solutions to that problem.
7
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Generalized Fokker-Planck equations and convergence to their equilibria

64%
Biler P., Karch G.
Banach Center Publications
|
2003
|
tom 60
|
nr 1
307-318
EN
We consider extensions of the classical Fokker-Planck equation uₜ + ℒu = ∇·(u∇V(x)) on $ℝ^{d}$ with ℒ = -Δ and V(x) = 1/2|x|², where ℒ is a general operator describing the diffusion and V is a suitable potential.
8
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Asymptotics for conservation laws involving Lévy diffusion generators

51%
Biler P., Karch G., Woyczyński W.
Studia Mathematica
|
2001
|
tom 148
|
nr 2
171-192
EN
Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their $L^{p}$-decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.
9
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Asymptotics for multifractal conservation laws

51%
Biler P., Karch G., Woyczynski W. A.
Studia Mathematica
|
1999
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tom 135
|
nr 3
231-252
EN
We study asymptotic behavior of solutions to multifractal Burgers-type equation $u_t + f(u)_x = Au$, where the operator A is a linear combination of fractional powers of the second derivative $-∂^2/ ∂ x^2$ and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the $L^p$-norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.
10
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Asymptotic stability of Navier-Stokes flow past an obstacle

51%
Biler P., Cannone M., Karch G.
Banach Center Publications
|
2004
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tom 66
|
nr 1
47-59
EN
Results on the asymptotic stability of solutions of the exterior Navier-Stokes problem in ℝ³ are proved in the framework of weak $L^p$ spaces.
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