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Displaying similar documents to “The analysis of blow-up solutions to a semilinear parabolic system with weighted localized terms”

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...

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 N , where Δ p is the p-Laplacian operator defined by Δ p u = d i v ( | u | p - 2 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.

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

Similarity:

We study the absence of nonnegative global solutions to parabolic inequalities of the type u t - ( - 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 ) a | x | - b , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling, Masahiko Shimojō (2019)

Mathematica Bohemica

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We consider solutions of quasilinear equations u t = Δ u m + u p in N with the initial data u 0 satisfying 0 < u 0 < M and lim | x | u 0 ( x ) = M for some constant M > 0 . It is known that if 0 < m < p with p > 1 , the blow-up set is empty. We find solutions u that blow up throughout N when m > p > 1 .

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

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

Similarity:

Consider the nonlinear heat equation (E): u t - Δ u = | u | p - 1 u + b | u | q . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C ( T - t ) - 1 / ( p - 1 ) | | u ( t ) | | C ( 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 u p . More general inequalities of the form u t - u x x 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...

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 2 ) semilinear parabolic equations u t = ( Δ ) m u ± | u | p 1 u in N × + ( 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 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.

Bi-spaces global attractors in abstract parabolic equations

J. W. Cholewa, T. Dłotko (2003)

Banach Center Publications

Similarity:

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. ...

Blow up for a completely coupled Fujita type reaction-diffusion system

Noureddine Igbida, Mokhtar Kirane (2002)

Colloquium Mathematicae

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This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form u - Δ ( a 11 u ) = h ( t , x ) | v | p , v - Δ ( a 21 u ) - Δ ( a 22 v ) = k ( t , x ) | w | q , w - Δ ( a 31 u ) - Δ ( a 32 v ) - Δ ( a 33 w ) = l ( t , x ) | u | r , for x N , t > 0, p > 0, q > 0, r > 0, a i j = a i j ( t , x , u , v ) , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for x N , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

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In this paper we examine self-similar solutions to the system u i t - d i Δ u i = k = 1 m u k p k i , i = 1,…,m, x N , t > 0, u i ( 0 , x ) = u 0 i ( x ) , i = 1,…,m, x N , where m > 1 and p k i > 0 , to describe asymptotics near the blow up point.

Isometric composition operators on weighted Dirichlet space

Shi-An Han, Ze-Hua Zhou (2016)

Czechoslovak Mathematical Journal

Similarity:

We investigate isometric composition operators on the weighted Dirichlet space 𝒟 α with standard weights ( 1 - | z | 2 ) α , α > - 1 . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space 𝒟 . We solve some of these but not in general. We also investigate the situation when 𝒟 α is equipped with another equivalent norm.

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 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 : = Ω , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

The regularity of the positive part of functions in L 2 ( I ; H 1 ( Ω ) ) H 1 ( I ; H 1 ( Ω ) * ) with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let u L 2 ( I ; H 1 ( Ω ) ) with t u 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 t u + 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.

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 ⎧ ( e β u - 1 ) / t - d i v ( | u | p - 2 u ) + d i v ( c ( x , t ) | u | s - 1 u ) + b ( x , t ) | 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 - 1 ) ( x ) in Ω. with s = (N+2)/(N+p) (p-1), c ( x , t ) ( L τ ( Q T ) ) N , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), b ( x , t ) L N + 2 , 1 ( Q T ) and f ∈ L¹(Q).

Logarithmically improved blow-up criterion for smooth solutions to the Leray- α -magnetohydrodynamic equations

Ines Ben Omrane, Sadek Gala, Jae-Myoung Kim, Maria Alessandra Ragusa (2019)

Archivum Mathematicum

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In this paper, the Cauchy problem for the 3 D Leray- α -MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray- α -MHD model in terms of the magnetic field B only in the framework of homogeneous Besov space with negative index.

Divergent solutions to the 5D Hartree equations

Daomin Cao, Qing Guo (2011)

Colloquium Mathematicae

Similarity:

We consider the Cauchy problem for the focusing Hartree equation i u t + Δ u + ( | · | - 3 | u | ² ) u = 0 in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of - Q + Δ Q + ( | · | - 3 | Q | ² ) Q = 0 in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂...

The structure of a local embedding and Chern classes of weighted blow-ups

Anca M. Mustaţǎ, Andrei Mustaţǎ (2012)

Journal of the European Mathematical Society

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For a proper local embedding between two Deligne-Mumford stacks Y and X , we find, under certain mild conditions, a new (possibly non-separated) Deligne-Mumford stack X ' , with an etale, surjective and universally closed map to the target X , and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to Y . Moreover, a natural set of weights on the substacks of X ' allows the construction of a universally...

Weighted H p spaces

José García-Cuerva

Similarity:

CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted H p spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u - χ · ( u v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω n , χ , r , μ > 0 , k > 1 and n 1 . It is shown with k ( 1 , 2 ) that the system possesses a global generalized solution for n 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1 .