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1
Content available remote

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

100%
Rencławowicz J.
Colloquium Mathematicum
|
2000
|
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 $p_k^i$.
2
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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
|
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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Self-similar solutions in reaction-diffusion systems

100%
Rencławowicz J.
Banach Center Publications
|
2003
|
tom 60
|
nr 1
337-346
EN
In this paper we examine self-similar solutions to the system $u_{it} - d_{i}Δu_{i} = ∏_{k=1}^{m} u^{p^{i}_{k}}_{k}$, i = 1,…,m, $x ∈ ℝ^{N}$, t > 0, $u_{i}(0,x) = u_{0i}(x)$, i = 1,…,m, $x ∈ ℝ^{N}$, where m > 1 and $p^{i}_{k} > 0$, to describe asymptotics near the blow up point.
4
Content available remote

Global existence and blow-up for a completely coupled Fujita type system

100%
Rencławowicz J.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 2
203-218
EN
The Fujita type global existence and blow-up theorems are proved for a reaction-diffusion system of m equations (m>1) in the form $u_{it} = Δu_i + u_{i+1}^{p_i}, i=1,..., m-1,$ $u_{mt} = Δu_m + u_1^{p_m}, x ∈ ℝ^N, t > 0,$ with nonnegative, bounded, continuous initial values and positive numbers $p_i$. The dependence on $p_i$ of the length of existence time (its finiteness or infinitude) is established.
5
Content available remote

Existence of solutions to the Poisson equation in $L_p$-weighted spaces

64%
Rencławowicz J., Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 1
1-12
EN
We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of $L_p$, p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.
6
Content available remote

Existence of solutions to the Poisson equation in L₂-weighted spaces

64%
Rencławowicz J., Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 3
309-323
EN
We consider the Poisson equation with the Dirichlet and the Neumann boundary conditions in weighted Sobolev spaces. The weight is a positive power of the distance to a distinguished plane. We prove the existence of solutions in a suitably defined weighted space.
7
Content available remote

Weak solutions to the Navier-Stokes equations in a Y-shaped domain

64%
Rencławowicz J., Zajączkowski W.
Applicationes Mathematicae
|
2006
|
tom 33
|
nr 1
111-127
EN
We prove the existence of weak solutions to the Navier-Stokes equations describing the motion of a fluid in a Y-shaped domain.
8
Content available remote

Local existence of solutions of the mixed problem for the system of equations of ideal relativistic hydrodynamics

64%
Rencławowicz J., Zajączkowski W. M.
Applicationes Mathematicae
|
1998-1999
|
tom 25
|
nr 2
221-252
EN
Existence and uniqueness of local solutions for the initial-boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Existence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich's mollifiers technique. Finally, existence for the nonlinear problem is proved by the method of successive approximations.
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