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  1. 13 Demonstratio Mathematica
  2. 6 Discussiones Mathematicae, Differential Inclusions, Control and Optimization
  3. 5 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
  4. 3 Open Mathematics
  5. 3 Opuscula Mathematica
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  1. 1 2021
  2. 1 2017
  3. 2 2016
  4. 2 2015
  5. 2 2014
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  1. 3 Povstenko Y.
  2. 3 Sikorska-Nowak A.
  3. 3 Szadkowska A.
  4. 2 Bochenek J.
  5. 2 Dlotko T.
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1
Content available remote The Cauchy problem for the nonlinear Schrödinger equations involving derivative terms in one spatial dimension
100%
Baoxiang W.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2003
|
tom [Z] 43(2)
275-300
EN
This paper is devoted to the study of the Cauchy problem for the nonlinear Schrödinger equations involving derivative terms. By introducing a generalized gauge transformation, we give some sufficient conditions for the global well-posedness of solutions in the energy space.
2
Content available remote A new kind of the solution of degenerate parabolic equation with unbounded convection term
80%
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.
3
Content available remote On the hierarchies of higher order mKdV and KdV equations
80%
Grünrock A.
Open Mathematics
|
2010
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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
Differential equations in banach space and henstock-kurzweil integrals
80%
Kubiaczyk I. , Sikorska A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1999
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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.
5
Content available remote On the classical solutions for parabolic differential - functional Cauchy problem
80%
Topolski K. A.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2004
|
tom Vol. 44, nr 2
217-226
EN
We consider the initial value problem for second order differential - functional parabolic equation. Functional dependence is of the Hale type. On the basis of differential inequalities and fixed point method we prove the existence theorem for classical solution. Our formulation covers a large group of nonlocal problems such as , integro-differential equations, and "retarded and deviated" argument. We put particular stress on the last one, as it requires more general treatment.
6
Content available On k-summability of formal solutions for certain partial differential operators with polynomial coefficients
80%
Ichinobe K. , Miyake M.
Opuscula Mathematica
|
2015
|
tom Vol. 35, no. 5
625--653
EN
We study the k-summability of divergent formal solutions for the Cauchy problem of certain linear partial differential operators with coefficients which are polynomial in t. We employ the method of successive approximation in order to construct the formal solutions and to obtain the properties of analytic continuation of the solutions of convolution equations and their exponential growth estimates.
7
Content available remote The Cauchy problem and self-similar solutions for a nonlinear parabolic equation
80%
Biler P.
Studia Mathematica
|
1995
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tom 114
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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.
8
Content available remote An existence theorem for solutions of an integro-differential equation in Banach spaces
80%
Dutkiewicz A.
Demonstratio Mathematica
|
2012
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tom Vol. 45, nr 4
895-899
EN
The paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.
9
Content available remote Second order evolution problem with dependent on t and not densely defined operators
80%
Winiarska T.
Demonstratio Mathematica
|
2009
|
tom Vol. 42, nr 3
513-520
EN
The purpose of this paper is to present some theorems on existence and uniqueness of solution for nonautonomous second order Cauchy problem with a dumping operator and with dependent on t not densely defined operators.
10
Content available remote The Cauchy problem for certain generalized diffrential equations of second order with singularity
80%
Szadkowska A.
Demonstratio Mathematica
|
2000
|
tom Vol. 33, nr 4
771-781
11
Content available remote Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals
80%
Sikorska-Nowak A.
Demonstratio Mathematica
|
2002
|
tom Vol. 35, nr 1
49-60
EN
In this paper, using properties of the Henstock-Kurzweil integral and corresponding theorems, we prove existence theorems for the equation x' = f(t,xt) and inclusion x' F(t,xt) in a Banach space where f is Henstock-Kurzweil integrable and satisfies some additional conditions.
12
Content available remote Parabolic problem in the space R∞
80%
Koroński J.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2004
|
tom Vol. 44, nr 1
93--98
EN
A Cauchy problem for homogeneous parabolic type partial differential equation in the infinite dimensional cartesian space is studied in this paper.
13
Content available remote Monotonic solutions of multi term fractional differential equations
80%
Salem H. A. H.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2007
|
tom Vol. 47, [Z] 1
1-9
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. Emmanuelle is the main tool in carring out our proof.
14
Content available remote Cauchy problem for quasilinear hyperbolic systems with coefficients functionally dependent on solutions
80%
Zdanowicz M. , Peradzyński Z.
Archives of Mechanics
|
2007
|
tom Vol. 59, nr 2
173-197
EN
The cauchy problem for a quasilinear hyperbolic systems with coefficients functionally dependent on the solutions is studied. We assume that the coefficients are continuous nonlinear operators in the Banach space C1 (R) satisfying some additional assumptions. Under these assumptions we prove the uniqueness and existence of local in time C1 solutions, provided that the initial data are also of class C1.
15
Content available remote An existence theorem for set differential inclusions in a semilinear metric space
80%
De Blasi F. S. , Lakshmikantham V. , Bhaskar T. G.
Control and Cybernetics
|
2007
|
tom Vol. 36, no 3
571-582
EN
Using the notion of continuous approximate selections, we establish an existence theorem for set differential inclusions in a semi-linear metric space.
16
Content available remote An abstract second order Cauchy problem with non-densely defined operator, I
80%
Bochenek J.
Demonstratio Mathematica
|
2000
|
tom Vol. 33, nr 3
489-501
EN
By using the theory of extrapolation space X-1 associated with an operator A which is non densely defined in Banach space X, the existence and uniqueness of solutions of linear second order differential initial value problem (1) is proved.
17
Content available Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation
80%
Povstenko Y. , Klekot J.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
tom Vol. 13, nr 1
95--102
EN
The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The numerical results are illustrated graphically.
18
Content available remote The Cauchy problem for certain generalized defferential equations of first order with singularity
80%
Szadkowska A.
Demonstratio Mathematica
|
1999
|
tom Vol. 32, nr 2
303-322
19
Content available remote Parabolic semilinear differential problem with nonlocal conditions in an unbounded domain
80%
Urbańska K.
Demonstratio Mathematica
|
2003
|
tom Vol. 36, nr 2
369--382
EN
This paper deals with the mixed problem for the semilinear parabolic equation of the second order in an unbounded domain with some nonlocal boundary data. We prove that there exists the unique global time solution for any locally integrable initial data and right-hand term: hence, no growth condition at infinity for these functions is required.
20
Content available remote On the existence of viscosity solutions for the differential-functional Cauchy problem
80%
Topolski K. A.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
1999
|
tom [Z] 39
207-223
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