Estimates of weak solutions to nondiagonal quasilinear parabolic systems

Dmitry Portnyagin

Displaying similar documents to “Estimates of weak solutions to nondiagonal quasilinear parabolic systems”

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.

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.

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. ...

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.

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.

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

P. Besala (1971)

Annales Polonici Mathematici

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

Eugene Fabes (1972)

Studia Mathematica

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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...

Boundary estimates for certain degenerate and singular parabolic equations

Benny Avelin, Ugo Gianazza, Sandro Salsa (2016)

Journal of the European Mathematical Society

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We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$ -Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_{T}$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of...

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....

Gradient estimates for the p(x)-Laplacian equation in $ℝ^{N}$

Chao Zhang, Shulin Zhou, Bin Ge (2015)

Annales Polonici Mathematici

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Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in $ℝ^{N}$ .

$L^{p} - L^{q}$ time decay estimates for the solution of the linear partial differential equations of thermodiffusion

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 $L^{p} - L^{q}$ time decay estimates for the solution of the associated linear Cauchy problem.

Strong maximum and minimum principles for parabolic functional-differential problems with nonlocal inequalities $[u^{j} (t_{0}, x) - K^{j}] + Σ_{i} h_{i} (x) [u^{j} (T_{i}, x) - K^{j}] \leq (\geq) 0$

Ludwik Byszewski (1990)

Annales Polonici Mathematici

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Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

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.

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang Phung, Gengsheng Wang (2013)

Journal of the European Mathematical Society

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This paper presents a new observability estimate for parabolic equations in $Ω \times (0, T)$ , where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0, T)$ . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

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 admissibility for parabolic equations in ℝⁿ

Martino Prizzi (2003)

Fundamenta Mathematicae

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We consider the parabolic equation (P) $u_{t} - Δ u = F (x, u)$ , (t,x) ∈ ℝ₊ × ℝⁿ, and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained...

$L^{p}$ -decay of solutions to dissipative-dispersive perturbations of conservation laws

Grzegorz Karch (1997)

Annales Polonici Mathematici

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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.

Strong maximum and minimum principles for parabolic functional-differential problems with initial inequalities $u (t_{0}, x) \leq (\geq) K$

Ludwik Byszewski (1990)

Annales Polonici Mathematici

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The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

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The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$ -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Existence result for nonlinear parabolic problems with L¹-data

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 $\partial u / \partial t - d i v [| D u - Θ (u) |^{p - 2} (D u - Θ (u))] + α (u) = f$ in ]0,T[ × Ω, $(| D u - Θ (u) |^{p - 2} (D u - Θ (u))) \cdot η + γ (u) = g$ 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.

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

On the fractional differentiability of the spatial derivatives of weak solutions to nonlinear parabolic systems of higher order

Roberto Amato (2016)

Czechoslovak Mathematical Journal

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We are concerned with the problem of differentiability of the derivatives of order $m + 1$ of solutions to the “nonlinear basic systems” of the type ${(- 1)}^{m} \sum_{| α | = m} D^{α} A^{α} (D^{(m)} u) + \frac{\partial u}{\partial t} = 0 in Q .$ We are able to show that $D^{α} u \in L^{2} (- a, 0, H^{ϑ} (B (σ), ℝ^{N})), | α | = m + 1,$ for $ϑ \in (0, 1 / 2)$ and this result suggests that more regularity is not expectable.

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...

Comparison theorems for infinite systems of parabolic functional-differential equations

Danuta Jaruszewska-Walczak (2001)

Annales Polonici Mathematici

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The paper deals with a weakly coupled system of functional-differential equations $\partial_{t} u_{i} (t, x) = f_{i} (t, x, u (t, x), u, \partial_{x} u_{i} (t, x), \partial_{x x} u_{i} (t, x))$ , i ∈ S, where (t,x) = (t,x₁,...,xₙ) ∈ (0,a) × G, $u = {u_{i}}_{i \in S}$ and S is an arbitrary set of indices. Initial boundary conditions are considered and the following questions are discussed: estimates of solutions, criteria of uniqueness, continuous dependence of solutions on given functions. The right hand sides of the equations satisfy nonlinear estimates of the Perron type with respect to the unknown functions. The...

The regularity of the positive part of functions in $L^{2} (I; H^{1} (Ω)) \cap H^{1} (I; H^{1} {(Ω)}^{*})$ with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $u \in L^{2} (I; H^{1} (Ω))$ with $\partial_{t} u \in L^{2} (I; H^{1} {(Ω)}^{*})$ be given. Then we show by means of a counter-example that the positive part $u^{+}$ of $u$ has less regularity, in particular it holds $\partial_{t} u^{+} \notin L^{1} (I; H^{1} {(Ω)}^{*})$ in general. Nevertheless, $u^{+}$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Adam Osękowski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $ℙ (s u p_{t \geq 0} | Y_{t} | \geq 1) \leq 3.375 . . . ∥ X ∥ ₁$ . As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

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. ...