Displaying similar documents to “Nonlinear evolution equations generated by subdifferentials with nonlocal constraints”

On discontinuous quasi-variational inequalities

Liang-Ju Chu, Ching-Yang Lin (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and T : X 2 X * and S : C 2 D are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a...

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....

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...

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).

On a semilinear variational problem

Bernd Schmidt (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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We provide a detailed analysis of the minimizers of the functional u n | u | 2 + D n | u | γ , γ ( 0 , 2 ) , subject to the constraint u L 2 = 1 . This problem,, describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative...

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. ...

Some results on quasi-t-dual Baer modules

Rachid Tribak, Yahya Talebi, Mehrab Hosseinpour (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a ring and let M be an R -module with S = End R ( M ) . Consider the preradical Z ¯ for the category of right R -modules Mod- R introduced by Y. Talebi and N. Vanaja in 2002 and defined by Z ¯ ( M ) = { U M : M / U is small in its injective hull } . The module M is called quasi-t-dual Baer if ϕ ϕ ( Z ¯ 2 ( M ) ) is a direct summand of M for every two-sided ideal of S , where Z ¯ 2 ( M ) = Z ¯ ( Z ¯ ( M ) ) . In this paper, we show that M is quasi-t-dual Baer if and only if Z ¯ 2 ( M ) is a direct summand of M and Z ¯ 2 ( M ) is a quasi-dual Baer module. It is also shown that any direct...

A penalty approach for a box constrained variational inequality problem

Zahira Kebaili, Djamel Benterki (2018)

Applications of Mathematics

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We propose a penalty approach for a box constrained variational inequality problem ( BVIP ) . This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of BVIP when the function F involved is continuous and strongly monotone and the box C contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results...

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.

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...

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

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We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ - d i v ( | x | - a p | u | p - 2 u ) + h ( x ) | u | p - 2 u = g ( x ) | u | r - 2 u , x ∈ Ω, ⎨ ⎩ | x | - a p | u | p - 2 u / ν = λ f ( x ) | u | q - 2 u , x ∈ ∂Ω, where Ω is an exterior domain in N , that is, Ω = N D , where D is a bounded domain in N with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function...

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.

Nonvariational basic parabolic systems of second order

Sergio Campanato (1991)

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

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Ω is a bounded open set of R n , of class C 2 and T > 0 . In the cylinder Q = Ω × 0 , T we consider non variational basic operator a H u - u / t where a ξ is a vector in R N , N 1 , which is continuous in ξ and satisfies the condition (A). It is shown that f L 2 Q the Cauchy-Dirichlet problem u W 0 2 , 1 Q , a H u - u / t = f in Q , has a unique solution. It is further shown that if u W 0 2 , 1 Q is a solution of the basic system a H u - u / t = 0 in Q , then H u and u / t belong to H l o c 1 Q . From this the Hölder continuity in Q of the vectors u and D u are deduced respectively when n 4 and...

Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski (2013)

Colloquium Mathematicae

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A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions f , . . . , f k : I , k ≥ 2, denoted by A [ f , . . . , f k ] , is considered. Some properties of A [ f , . . . , f k ] , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For...

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

Liguang Liu, Dachun Yang (2009)

Studia Mathematica

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Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space p , q s ( ) to a quasi-Banach space ℬ if and only if sup | | T ( a ) | | : a is an infinitely differentiable (p,q,s)-atom of p , q s ( ) < ∞, where the (p,q,s)-atom of p , q s ( ) is as defined by Han, Paluszyński and Weiss.