Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

Gabriel Acosta; Julián Fernández Bonder; Pablo Groisman; Julio Daniel Rossi

Displaying similar documents to “Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions”

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...

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 >...

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$ : either the space derivative $u_{x}$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^{1}$ norm to the unique steady state. The main difficulty is to prove $C^{1}$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out...

Blow-up of solutions for the non-Newtonian polytropic filtration equation with a generalized source

Jun Zhou (2016)

Annales Polonici Mathematici

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This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation $u_{t} - d i v (| \nabla u^{m} |^{p - 2} \nabla u^{m}) = f (u)$ with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.

Blow-up of solutions for the Kirchhoff equation of q-Laplacian type with nonlinear dissipation

Abbes Benaissa, Salim A. Messaoudi (2002)

Colloquium Mathematicae

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We establish the blow-up of solutions to the Kirchhoff equation of q-Laplacian type with a nonlinear dissipative term $(| u_{t} |^{l - 2} u_{t})_{t} - M (| | A^{1 / 2} u | | ² ₂) A u + | u_{t} |^{β} u_{t} = {| u |}^{p} u$ , x ∈ Ω, t > 0.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan Martel, Frank Merle, Pierre Raphaël (2015)

Journal of the European Mathematical Society

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We consider the mass critical (gKdV) equation $u_{t} + {(u_{x x} + u^{5})}_{x} = 0$ for initial data in $H^{1}$ . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

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We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system $Δ_{p} u = g (u - α v), Δ_{p} v = f (v - β u)$ in a smooth bounded domain of $ℝ^{N}$ , where $Δ_{p}$ is the p-Laplacian operator defined by $Δ_{p} u = {d i v (| \nabla u |}^{p - 2} \nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2011)

Journal of the European Mathematical Society

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Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let $W$ be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially...

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.

Numerical solution of parabolic equations in high dimensions

Tobias Von Petersdorff, Christoph Schwab (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We consider the numerical solution of diffusion problems in $(0, T) \times Ω$ for $Ω \subset ℝ^{d}$ and for $T > 0$ in dimension $d \geq 1$ . We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p \geq 1$ , and $h p$ discontinuous Galerkin time-discretization of order $r = O (|log h|)$ on a geometric sequence of $O (|log h|)$ many time steps. The linear systems in each time step are solved iteratively by $O (|log h|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an $L^{2} (Ω)$ -error of $O (N^{- p})$ for $u (x, T)$ where $N$ is the total number of...

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.

Div-curl lemma revisited: Applications in electromagnetism

Marián Slodička, Ján Jr. Buša (2010)

Kybernetika

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Two new time-dependent versions of div-curl results in a bounded domain $Ω \subset ℝ^{3}$ are presented. We study a limit of the product $v_{k} w_{k}$ , where the sequences $v_{k}$ and $w_{k}$ belong to $Ł_{2} (Ω)$ . In Theorem 2.1 we assume that $\nabla \times v_{k}$ is bounded in the $L_{p}$ -norm and $\nabla \cdot w_{k}$ is controlled in the $L_{r}$ -norm. In Theorem 2.2 we suppose that $\nabla \times w_{k}$ is bounded in the $L_{p}$ -norm and $\nabla \cdot w_{k}$ is controlled in the $L_{r}$ -norm. The time derivative of $w_{k}$ is bounded in both cases in the norm of $- 1 (Ω)$ . The convergence (in the sense of distributions) of $v_{k} w_{k}$ to the product $v w$ ...

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....

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...

Numerical blow-up for the $p$ -Laplacian equation with a nonlinear source

Ferreira, Raúl, de Pablo, Arturo, Pérez-Llanos, Mayte

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