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

P. Besala

Displaying similar documents to “On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations”

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...

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).

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.

On higher-order semilinear parabolic equations with measures as initial data

Victor Galaktionov (2004)

Journal of the European Mathematical Society

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We consider $2 m$ th-order $(m \geq 2)$ semilinear parabolic equations $u_{t} = - {(- Δ)}^{m} u \pm {| u |}^{p - 1} u$ in $ℛ^{N} \times ℝ_{+}$ $(p > 1)$ , with Dirac’s mass $δ (x)$ as the initial function. We show that for $p < p_{0} = 1 + 2 m / N$ , the Cauchy problem admits a solution $u (x, t)$ which is bounded and smooth for small $t > 0$ , while for $p \geq p_{0}$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

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....

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. ...

$L^{p} - L^{q}$ 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.

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

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It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_{t} (t) + A u (t) = f (t, u (t))$ , $0 \leq t < τ$ with $u (τ) = φ$ , where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $φ$ and $f$ , that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$ , which is also ill-posed. Spectral truncation...

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.

Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems

Tayeb Benhamoud, Elmehdi Zaouche, Mahmoud Bousselsal (2024)

Mathematica Bohemica

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This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_{t} - M (\int_{Ω} φ u d x) div (A (x, t, u) \nabla u) = g (x, t, u)$ in $Ω \times (0, T)$ , where $Ω$ is a bounded domain of $ℝ^{n}$ $(n \geq 1)$ , $T > 0$ is a positive number, $A (x, t, u)$ is an $n \times n$ matrix of variable coefficients depending on $u$ and $M : ℝ \to ℝ$ , $φ : Ω \to ℝ$ , $g : Ω \times (0, T) \times ℝ \to ℝ$ are given functions. We consider two different assumptions on $g$ . The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A (x, t, u) = a (x, t)$ depends only on...

Estimates of $\sum_{k = 1}^{N} k^{- s} 〈 k x 〉^{- t}$

A. Kruse (1967)

Acta Arithmetica

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Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial_{t} u + {div}_{y} A (y, u) - Δ_{y} u = 0$ , where $y \in ℝ^{N}$ and the flux $A$ is periodic in $y$ . More specifically, we consider the case when the initial data is an $L^{1}$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^{1}$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

On a new normalization for tractor covariant derivatives

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)

Journal of the European Mathematical Society

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A regular normal parabolic geometry of type $G / P$ on a manifold $M$ gives rise to sequences $D_{i}$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\nabla^{ω}$ on the corresponding tractor bundle $V$ , where $ω$ is the normal Cartan connection. The first operator $D_{0}$ in the sequence is overdetermined and it is well known that $\nabla^{ω}$ yields the prolongation of this operator in the homogeneous case $M = G / P$ . Our first...

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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On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Malkhaz Ashordia (2016)

Mathematica Bohemica

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The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $d x (t) = d A_{0} (t) \cdot x (t) + d f_{0} (t)$ , $x (t_{0}) = c_{0}$ $(t \in I)$ with a unique solution $x_{0}$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $d x (t) = d A_{k} (t) \cdot x (t) + d f_{k} (t)$ , $x (t_{k}) = c_{k}$ $(k = 1, 2, \dots)$ to have a unique solution $x_{k}$ for any sufficiently large $k$ such that $x_{k} (t) \to x_{0} (t)$ uniformly on $I$ . Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

Vectorial quasilinear diffusion equation with dynamic boundary condition

Nakayashiki, Ryota

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In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S) $_{ε}$ with a nonnegative constant $ε$ , and for any $ε \geq 0$ , (S) $_{ε}$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $Ω$ , and the parabolic equation on the boundary $Γ : = \partial Ω$ , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ${ε | u |}^{α - 1} u$ and ${δ | b |}^{β - 1} b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

Porous Medium Type Equations with a Quadratic Gradient Term

Daniela Giachetti, Giulia Maroscia (2007)

Bollettino dell'Unione Matematica Italiana

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We show an existence result for the Cauchy-Dirichlet problem in $Q_{T}=\Omega\times(0,T)$ for parabolic equations with degenerate principal part (of porous medium type) with a lower order term having a quadratic growth with respect to the gradient. The right hand side of the equation $f$ and the initial datum $u_{0}$ are bounded nonnegative functions.

Displaying similar documents to “On L p -estimates for solutions of the Cauchy problem for parabolic differential equations”

Displaying similar documents to “On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations”