Displaying similar documents to “Existence results for a class of nonlinear parabolic equations with two lower order terms”

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 | 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...

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, 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 ( Ω φ u d x ) div ( A ( x , t , u ) u ) = g ( x , t , u ) in Ω × ( 0 , T ) , where Ω is a bounded domain of n ( n 1 ) , T > 0 is a positive number, A ( x , t , u ) is an n × n matrix of variable coefficients depending on u and M : , φ : Ω , g : Ω × ( 0 , T ) × 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...

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.

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 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...

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,...

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 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 ω yields the prolongation of this operator in the homogeneous case M = G / P . Our first...

Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)

Journal of the European Mathematical Society

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In this work we study the problem u t div ( | x | 2 γ u ) = λ u α | x | 2 ( γ + 1 ) + f in Ω × ( 0 , T ) , u 0 in Ω × ( 0 , T ) , u = 0 on Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , Ω N ( N 2 ) is a bounded regular domain such that 0 Ω , λ > 0 , α > 0 , - < γ < ( N 2 ) / 2 , f and u 0 are positive functions such that f L 1 ( Ω × ( 0 , T ) ) and u 0 L 1 ( Ω ) . The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case α = 1 , 1 + γ > 0 ; (ii) the nonexistence of solutions if α > 1 , 1 + γ > 0 ; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values u ( η 1 , t ) , , u ( η q , t ) with 0 η 1 < η 2 < < η q < 1 . By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case ( P q ) of (P) in which the nonlinear term contains the sum S q [ u 2 ] ( t ) = q - 1 i = 1 q u 2 ( ( i - 1 ) q , t ) . Under suitable conditions, we prove that the solution of ( P q ) converges to the solution...

Perturbed nonlinear degenerate problems in N

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ d i v ( x , u ) + a ( x ) | u | p - 2 u = g ( x ) | u | p - 2 u + h ( x ) | u | s - 1 u in N ⎨ ⎩ u > 0, l i m | x | u ( x ) = 0 , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth

Mario Marino, Antonino Maugeri (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Sfruttando i risultati di [1], si prova che le derivate spaziali D α u di ordine | α | con | α | < m - 1 delle soluzioni in Q di un sistema parabolico quasilineare di ordine 2 m con andamenti strettamente controllati, sono parzialmente hölderiane in Q con esponente di hölderianità decrescente al crescere di | α | .

Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form

Mohamed Saad Bouh Elemine Vall, Ahmed Ahmed, Abdelfattah Touzani, Abdelmoujib Benkirane (2018)

Mathematica Bohemica

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We prove the existence of solutions to nonlinear parabolic problems of the following type: b ( u ) t + A ( u ) = f + div ( Θ ( x ; t ; u ) ) in Q , u ( x ; t ) = 0 on Ω × [ 0 ; T ] , b ( u ) ( t = 0 ) = b ( u 0 ) on Ω , where b : is a strictly increasing function of class 𝒞 1 , the term A ( u ) = - div ( a ( x , t , u , u ) ) is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, Θ : Ω × [ 0 ; T ] × is a Carathéodory, noncoercive function which satisfies the following condition: sup | s | k | Θ ( · , · , s ) | E ψ ( Q ) for all k > 0 , where ψ is the Musielak complementary function of Θ , and the second term f belongs to L 1 ( Q ) .

Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space

Michael Gil&#039; (2023)

Czechoslovak Mathematical Journal

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We consider the equation d y ( t ) / d t = ( A + B ( t ) ) y ( t ) ( t 0 ) , where A is the generator of an analytic semigroup ( e A t ) t 0 on a Banach space 𝒳 , B ( t ) is a variable bounded operator in 𝒳 . It is assumed that the commutator K ( t ) = A B ( t ) - B ( t ) A has the following property: there is a linear operator S having a bounded left-inverse operator S l - 1 such that S e A t is integrable and the operator K ( t ) S l - 1 is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

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 .

Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, K. P. Rybakowski (2002)

Studia Mathematica

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Consider the family uₜ = Δu + G(u), t > 0, x Ω ε , ν ε u = 0 , t > 0, x Ω ε , ( E ε ) of semilinear Neumann boundary value problems, where, for ε > 0 small, the set Ω ε is a thin domain in l , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of l . If G is dissipative, then equation ( E ε ) has a global attractor ε . We identify a “limit” equation for the family ( E ε ) , prove convergence of trajectories and establish an upper semicontinuity result for the family ε as ε → 0⁺. ...

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 t u + div y A ( y , u ) - Δ y u = 0 , where y 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...

A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type

Soohyun Bae (2023)

Archivum Mathematicum

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We consider the quasilinear equation Δ p u + K ( | x | ) u q = 0 , and present the proof of the local existence of positive radial solutions near 0 under suitable conditions on K . Moreover, we provide a priori estimates of positive radial solutions near when r - K ( r ) for - p is bounded near .

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 - ( - 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....

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N - β , with β [ 0 , ) . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β . If β [ 0 , 1 ) , the hydrodynamic limit is given by the usual heat equation. If β = 1 , it is given by a parabolic equation involving an operator...

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( P λ ) ⎩ u Ω = 0 where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( P λ ) admits a non-zero, non-negative strong solution u λ p 2 W 2 , p ( Ω ) such that l i m λ 0 | | u λ | | W 2 , p ( Ω ) = 0 for all p ≥ 2. Moreover, the function λ I λ ( u λ ) is negative and decreasing in ]0,λ*[, where I λ is the energy functional related to ( P λ ). ...

Partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth

Mario Marino, Antonino Maugeri (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Sfruttando i risultati di [1], si prova che le derivate spaziali D α u di ordine | α | con | α | < m - 1 delle soluzioni in Q di un sistema parabolico quasilineare di ordine 2 m con andamenti strettamente controllati, sono parzialmente hölderiane in Q con esponente di hölderianità decrescente al crescere di | α | .