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1
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The Cauchy problem and self-similar solutions for a nonlinear parabolic equation

100%
Biler P.
Studia Mathematica
|
1995
|
tom 114
|
nr 2
181-205
EN
The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.
2

Differential equations in banach space and henstock-kurzweil integrals

100%
Kubiaczyk I., Sikorska A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1999
|
tom 19
|
nr 1-2
35-43
EN
In this paper, using the properties of the Henstock-Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x' = f(t,x) and inclusion x' ∈ F(t,x) in a Banach space, where f is Henstock-Kurzweil integrable and satisfies some conditions.
3
Content available remote

On the hierarchies of higher order mKdV and KdV equations

100%
Grünrock A.
Open Mathematics
|
2010
|
tom 8
|
nr 3
500-536
EN
The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $$ \hat H_s^r \left( \mathbb{R} \right) $$ defined by the norm $$ \left\| {v_0 } \right\|_{\hat H_s^r \left( \mathbb{R} \right)} : = \left\| {\left\langle \xi \right\rangle ^s \widehat{v_0 }} \right\|_{L_\xi ^{r'} } , \left\langle \xi \right\rangle = \left( {1 + \xi ^2 } \right)^{\frac{1} {2}} , \frac{1} {r} + \frac{1} {{r'}} = 1 $$. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $$ \frac{{2j - 1}} {{2r'}} $$. The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $$ \frac{{2j - 1}} {{2r'}} $$. The results for r = 2 - so far in the literature only if j = 1 (mKdV) or j = 2 - can be combined with the higher order conservation laws for the mKdV equation to obtain global well-posedness of the jth equation in H s(ℝ) for s ≥ $$ \frac{{j + 1}} {2} $$, if j is odd, and for s ≥ $$ \frac{j} {2} $$, if j is even. - The Cauchy problem for the jth equation in the KdV hierarchy with data in $$ \hat H_s^r \left( \mathbb{R} \right) $$ cannot be solved by Picard iteration, if r > $$ \frac{{2j}} {{2j - 1}} $$, independent of the size of s ∈ ℝ. Especially for j ≥ 2 we have C 2-ill-posedness in H s(ℝ). With similar arguments as used before in the mKdV context it is shown that this problem is locally well-posed in $$ \hat H_s^r \left( \mathbb{R} \right) $$, if 1 < r ≤ $$ \frac{{2j}} {{2j - 1}} $$ and $$ s > j - \frac{3} {2} - \frac{1} {{2j}} + \frac{{2j - 1}} {{2r'}} $$. For KdV itself the lower bound on s is pushed further down to $$ s > max\left( { - \frac{1} {2} - \frac{1} {{2r'}} - \frac{1} {4} - \frac{{11}} {{8r'}}} \right) $$, where r ∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.
4
Content available remote

A new kind of the solution of degenerate parabolic equation with unbounded convection term

100%
Zhan H. Z.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
A new kind of entropy solution of Cauchy problem of the strong degenerate parabolic equation [...] is introduced. If u0 ∈ L∞(RN), E = {Ei} ∈ (L2(QT))N and divE ∈ L2(QT), by a modified regularization method and choosing the suitable test functions, the BV estimates are got, the existence of the entropy solution is obtained. At last, by Kruzkov bi-variables method, the stability of the solutions is obtained.
5

Weak solutions of differential equations in Banach spaces

100%
Cichoń M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1995
|
tom 15
|
nr 1
5-14
EN
In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.
6
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Monotonic solutions of multi term fractional differential equations

100%
Salem H. A. H.
Commentationes Mathematicae
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2007
|
tom 47
|
nr 1
EN
In this paper, we present an existence of monotonic solutions for a nonlinear multi term non-autonomous fractional differential equation in the Banach space of summable functions. The concept of measure of noncompactness and a fixed point theorem due to \(G\). Emmanuele is the main tool in carring out our proof.
7
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$L^p$-$L^q$-Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

88%
Gawinecki J.
Annales Polonici Mathematici
|
1991
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tom 54
|
nr 2
135-145
EN
We prove the $L^p$-$L^q$-time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman's approach [12], [5] to the $L^p$-$L^q$-time decay estimates.
8
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Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals

88%
Sikorska-Nowak A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2007
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tom 27
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nr 2
315-327
EN
We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.
9
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Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

88%
Litovchenko V., Dovzhytska I.
Open Mathematics
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2012
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tom 10
|
nr 3
1084-1102
EN
In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.
10
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Abstract parabolic problem with non-Lipschitz nonlinearity

88%
Cholewa J. W., Dlotko T.
Banach Center Publications
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2000
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tom 52
|
nr 1
73-81
EN
An abstract parabolic equation with sectorial operator and continuous nonlinearity is studied in this paper. In particular, the asymptotic behavior of solutions is described within the framework of the theory of global attractors. Examples included in the final part of the paper illustrate the presented ideas.
11
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Multivalued linear operators and differential inclusions in Banach spaces

75%
Baskakov A., Obukhovskii V., Zecca P.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2003
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tom 23
|
nr 1
53-74
EN
In this paper, we study multivalued linear operators (MLO's) and their resolvents in non reflexive Banach spaces, introducing a new condition of a minimal growth at infinity, more general than the Hille-Yosida condition. Then we describe the generalized semigroups induced by MLO's. We present a criterion for an MLO to be a generator of a generalized semigroup in an arbitrary Banach space. Finally, we obtain some existence results for differential inclusions with MLO's and various types of multivalued nonlinearities. As a consequence, we give theorems on the existence of local, global and bounded solutions of the Cauchy problem for degenerate differential inclusions.
12
Content available remote

Existence of the fundamental solution of a second order evolution equation

75%
Bochenek J.
Annales Polonici Mathematici
|
1997
|
tom 66
|
nr 1
15-35
EN
We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence ${A_n(t)}$ of bounded operators.
13
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An existence result for impulsive functional differential inclusions in Banach spaces

75%
Benedetti I.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2004
|
tom 24
|
nr 1
13-30
EN
We use the topological degree theory for condensing multimaps to present an existence result for impulsive semilinear functional differential inclusions in Banach spaces. Moreover, under some additional assumptions we prove the compactness of the solution set.
14
Content available remote

Tauberian theorems for vector-valued Fourier and Laplace transforms

75%
Chill R.
Studia Mathematica
|
1998
|
tom 128
|
nr 1
55-69
EN
Let X be a Banach space and $f ∈ L^1_loc(ℝ;X)$ be absolutely regular (i.e. integrable when divided by some polynomial). If the distributional Fourier transform of f is locally integrable then f converges to 0 at infinity in some sense to be made precise. From this result we deduce some Tauberian theorems for Fourier and Laplace transforms, which can be improved if the underlying Banach space has the analytic Radon-Nikodym property.
15
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An abstract Cauchy problem for higher order functional differential inclusions with infinite delay

63%
Ke T., Obukhovskii V., Wong N.-C., Yao J.-C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2011
|
tom 31
|
nr 2
199-229
EN
The existence results for an abstract Cauchy problem involving a higher order differential inclusion with infinite delay in a Banach space are obtained. We use the concept of the existence family to express the mild solutions and impose the suitable conditions on the nonlinearity via the measure of noncompactness in order to apply the theory of condensing multimaps for the demonstration of our results. An application to some classes of partial differential equations is given.
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