scholarly journals On some semilinear equations of Schrödinger type

2002 ◽  
Vol 90 (1) ◽  
pp. 139
Author(s):  
Rossella Agliardi ◽  
Daniela Mari
Keyword(s):  
Differential Equations ◽  
Existence Theorem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Local Existence ◽  
Weighted Sobolev Spaces ◽  
Initial Value ◽  
Semilinear Equations ◽  
Local Existence Theorem

We study the initial value problem for some semilinear pseudo-differential equations of the form $\partial _tu + i\ H(x,D_x)u = F(u,\nabla u)$. The assumptions we make on $H$ are trivially satisfied by $\Delta$, thus our equations generalize Schrdinger type equations. A local existence theorem is proved in some weighted Sobolev spaces.

scholarly journals A Singular Initial-Value Problem for Second-Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Second Order ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Local Existence And Uniqueness ◽  
Singular Initial Value Problem

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

AN EFFECTIVE CAUCHY-PEANO EXISTENCE THEOREM FOR UNIQUE SOLUTIONS

1996 ◽  
Vol 07 (02) ◽  
pp. 151-160 ◽  
Author(s):  
KEIJO RUOHONEN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Initial Value Problem ◽  
Uniqueness Of Solution ◽  
Time Interval ◽  
Initial Value ◽  
Initial Values ◽  
Systems Of Odes

It is shown in this paper that the solution of the initial value problem for a system of ordinary differential equations is computable if the following assumptions are satisfied: The time interval considered is computable, the system is continuous and computable, the initial values are computable, the system is effectively bounded, and the solution is unique. It should be mentioned that for a single ODE this follows immediately from the standard proof of Osgood’s existence theorem, but this approach is not available for systems of ODEs. The key assumption here is uniqueness of solution: a result of Pour-El’s and Richards’ shows that nonunique solutions may be noncomputable, even for a single ODE.

scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

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