Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

Kentarou Fujie; Tomomi Yokota

Displaying similar documents to “Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity”

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

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 $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

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 $ℝ^{N}$ : ∂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. ...

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

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We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . 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 $p = q$ . 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...

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} \geq - {(- Δ)}^{β / 2} u - V (x) u + h (x, t) u^{p}$ , where ${(- Δ)}^{β / 2}$ , 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 ${V ₊ (x) \sim a | x |}^{- b}$ , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$ -initial data

Daisuke Shimotsuma, Tomomi Yokota, Kentarou Yoshii (2014)

Mathematica Bohemica

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This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $\frac{\partial u}{\partial t} - (λ + i α) Δ u + (κ + i β) {| u |}^{q - 1} u - γ u = 0$ in $ℝ^{N} \times (0, \infty)$ with $L^{p}$ -initial data $u_{0}$ in the subcritical case ( $1 \leq q < 1 + 2 p / N$ ), where $u$ is a complex-valued unknown function, $α$ , $β$ , $γ$ , $κ \in ℝ$ , $λ > 0$ , $p > 1$ , $i = \sqrt{- 1}$ and $N \in ℕ$ . The proof is based on the $L^{p}$ - $L^{q}$ estimates of the linear semigroup ${exp (t (λ + i α) Δ)}$ and usual fixed-point argument.

On the long-time behaviour of solutions of the p-Laplacian parabolic system

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 $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Yūki Naito (2006)

Banach Center Publications

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We consider a nonlinear parabolic system modelling chemotaxis $u_{t} = \nabla \cdot (\nabla u - u \nabla v)$ , $v_{t} = Δ v + u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

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In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the...

$L^{\infty}$ estimates of solution for $m$ -Laplacian parabolic equation with a nonlocal term

Pulun Hou, Caisheng Chen (2011)

Czechoslovak Mathematical Journal

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In this paper, we consider the global existence, uniqueness and $L^{\infty}$ estimates of weak solutions to quasilinear parabolic equation of $m$ -Laplacian type $u_{t} - {div (| \nabla u |}^{m - 2} {\nabla u) = u | u |}^{β - 1} \int_{Ω} {| u |}^{α} d x$ in $Ω \times (0, \infty)$ with zero Dirichlet boundary condition in $\partial Ω$ . Further, we obtain the $L^{\infty}$ estimate of the solution $u (t)$ and $\nabla u (t)$ for $t > 0$ with the initial data $u_{0} \in L^{q} (Ω)$ $(q > 1)$ , and the case $α + β < m - 1$ .

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): $u_{t} - Δ u = {| u |}^{p - 1} u + b {| \nabla u |}^{q}$ . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C ₁ {(T - t)}^{- 1 / (p - 1)} {\leq | | u (t) | |}_{\infty} \leq C ₂ {(T - t)}^{- 1 / (p - 1)}$ . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{x x} \geq u^{p}$ . More general inequalities of the form $u_{t} - u_{x x} \geq f (u)$ with, for instance, $f (u) = (1 + u) l o g^{p} (1 + u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

Parabolic Marcinkiewicz integrals on product spaces and extrapolation

Mohammed Ali, Mohammed Al-Dolat (2016)

Open Mathematics

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In this paper, we study the the parabolic Marcinkiewicz integral [...] MΩ,hρ1,ρ2 $ℳ_{Ω, h}^{ρ_{1,} ρ_{2}}$ on product domains Rn × Rm (n, m ≥ 2). Lp estimates of such operators are obtained under weak conditions on the kernels. These estimates allow us to use an extrapolation argument to obtain some new and improved results on parabolic Marcinkiewicz integral operators.

The initial value problem for parabolic equations with data in $L^{p} (R^{n})$

Eugene Fabes (1972)

Studia Mathematica

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The analysis of blow-up solutions to a semilinear parabolic system with weighted localized terms

Haihua Lu, Feng Wang, Qiaoyun Jiang (2011)

Annales Polonici Mathematici

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This paper deals with blow-up properties of solutions to a semilinear parabolic system with weighted localized terms, subject to the homogeneous Dirichlet boundary conditions. We investigate the influence of the three factors: localized sources $u^{p} (x ₀, t)$ , vⁿ(x₀,t), local sources $u^{m} (x, t)$ , $v^{q} (x, t)$ , and weight functions a(x),b(x), on the asymptotic behavior of solutions. We obtain the uniform blow-up profiles not only for the cases m,q ≤ 1 or m,q > 1, but also for m > 1 q < 1 or m < 1 q >...

Existence results for a class of nonlinear parabolic equations with two lower order terms

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 ⎧ $\partial (e^{β u} - 1) / \partial t - {d i v (| \nabla u |}^{p - 2} \nabla u) + d i v (c (x, t) {| u |}^{s - 1} u) + b (x, t) {| \nabla u |}^{r} = f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{β u} - 1) (x, 0) = (e^{β u ₀} - 1) (x)$ in Ω. with s = (N+2)/(N+p) (p-1), $c (x, t) \in {(L^{τ} (Q T))}^{N}$ , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b (x, t) \in L^{N + 2, 1} (Q T)$ and f ∈ L¹(Q).

On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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Weak- $L^{p}$ solutions for a model of self-gravitating particles with an external potential

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- $L^{p}$ spaces (i.e. Markiewicz spaces). The main goal is to prove the existence of global solutions and to study their large time behaviour.

Estimates of Green functions and their applications for parabolic operators with singular potentials

Lotfi Riahi (2003)

Colloquium Mathematicae

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We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1, 1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

Bi-spaces global attractors in abstract parabolic equations

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 $X^{α}$ . This semigroup possesses an $(X^{α} - Z)$ -global attractor that is closed, bounded, invariant in $X^{α}$ , and attracts bounded subsets of $X^{α}$ 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. ...