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1
Content available remote

On nonstationary motion of a fixed mass of a viscous compressible barotropic fluid bounded by a free boundary

100%
Zadrzyńska E., Zajączkowski W. M.
Colloquium Mathematicum
|
1999
|
tom 79
|
nr 2
283-310
2
Content available remote

Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

100%
Rencławowicz J.
Applicationes Mathematicae
|
1998-1999
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tom 25
|
nr 3
313-326
EN
We examine the parabolic system of three equations $u_t$ - Δu = $v^p$, $v_t$ - Δv = $w^q$, $w_t$ - Δw = $u^r$, x ∈ $ℝ^N$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.
3
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On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid

100%
Zadrzyńska E., Zajączkowski W.
Annales Polonici Mathematici
|
1996
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tom 63
|
nr 3
199-221
EN
We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.
4
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Behaviour of global solutions for a system of reaction-diffusion equations from combustion theory

100%
Badraoui S.
Applicationes Mathematicae
|
1999
|
tom 26
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nr 2
133-150
EN
We are concerned with the boundedness and large time behaviour of the solution for a system of reaction-diffusion equations modelling complex consecutive reactions on a bounded domain under homogeneous Neumann boundary conditions. Using the techniques of E. Conway, D. Hoff and J. Smoller [3] we also show that the bounded solution converges to a constant function as t → ∞. Finally, we investigate the rate of this convergence.
5
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On global motion of a compressible barotropic viscous fluid with boundary slip condition

100%
Kobayashi T., Zajączkowski W. M.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 2
159-194
EN
Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ $ℝ^3$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the $L_2$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to $H^{2+α,1+α/2}(Ω × ℝ_+)$ and the density belongs to $H^{1+α,1/2+α/2}(Ω× ℝ_+)$, α ∈ (1/2,1).
6
Content available remote

Global pointwise a priori bounds and large time behaviour for a nonlinear system describing the spread of infectious disease

100%
Kirane M.
Applicationes Mathematicae
|
1993-1995
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tom 22
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nr 1
1-9
EN
This paper considers a reaction-diffusion system with biatic diffusion.Existence of a globally bounded solution is proved and its large timebehaviour is given.
7
Content available remote

Neumann problem for one-dimensional nonlinear thermoelasticity

100%
Shibata Y.
Banach Center Publications
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1992
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tom 27
|
nr 2
457-480
EN
The global existence theorem of classical solutions for one-dimensional nonlinear thermoelasticity is proved for small and smooth initial data in the case of a bounded reference configuration for a homogeneous medium, considering the Neumann type boundary conditions: traction free and insulated. Moreover, the asymptotic behaviour of solutions is investigated.
8
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On nonstationary motion of a fixed mass of a general viscous compressible heat conducting capillary fluid bounded by a free boundary

100%
Zadrzyńska E.
Applicationes Mathematicae
|
1998-1999
|
tom 25
|
nr 4
489-511
EN
The motion of a fixed mass of a viscous compressible heat conducting capillary fluid is examined. Assuming that the initial data are sufficiently close to a constant state and the external force vanishes we prove the existence of a global-in-time solution which is close to the constant state for any moment of time. Moreover, we present an analogous result for the case of a barotropic viscous compressible fluid.
9
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On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface

90%
Zajączkowski W. M.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We consider the motion of a viscous compressible barotropic fluid in $ℝ^3$ bounded by a free surface which is under constant exterior pressure. For a given initial density, initial domain and initial velocity we prove the existence of local-in-time highly regular solutions. Next assuming that the initial density is sufficiently close to a constant, the initial pressure is sufficiently close to the external pressure, the initial velocity is sufficiently small and the external force vanishes we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.
XX
CONTENTS 1. Introduction.......................................5 2. Global estimates and relations........11 3. Local existence...............................16 4. Global differential inequality............44 5. Korn inequality................................81 6. Global existence.............................89 References.......................................100
10
Content available remote

Blow up, global existence and growth rate estimates in nonlinear parabolic systems

88%
Rencławowicz J.
Colloquium Mathematicum
|
2000
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tom 86
|
nr 1
43-66
EN
We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. $u_{it} - d_{i} Δu_{i} = \prod_{k=1}^m u_{k}^{p_k^i}, i=1,...,m, x ∈ ℝ^{N}, t > 0,$ with nonnegative, bounded, continuous initial values and $p_{k}^{i} ≥ 0$, $i,k = 1,...,m$, $d_i > 0$, $i = 1,...,m$. For solutions which blow up at $t = T <≤ ∞$, we derive the following bounds on the blow up rate: $u_i(x,t) ≤ C(T - t)^{-α_{i}}$ with C > 0 and $α_i$ defined in terms of $p_k^i$.
11
Content available remote

Global solutions via partial information and the Cahn-Hilliard equation

88%
Cholewa J. W., Dłotko T.
Banach Center Publications
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1996
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tom 33
|
nr 1
39-50
EN
Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.
12
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Global existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditions

88%
Nouioua F., Basti B.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2021
|
tom 20
43-56
EN
This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder's and Banach's fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.
13
Content available remote

On quasistatic inelastic models of gradient type with convex composite constitutive equations

63%
Chełmiński K.
Open Mathematics
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2003
|
tom 1
|
nr 4
670-689
EN
This article defines and presents the mathematical analysis of a new class of models from the theory of inelastic deformations of metals. This new class, containing so called convex composite models, enlarges the class containing monotone models of gradient type defined in [1]. This paper starts to establish the existence theory for models from this new class; we restrict our investigations to the coercive and linear self-controlling case.
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