Dissertation > Excellent graduate degree dissertation topics show

Existence of Solutions for a Class of Nonlinear Equations with L~1 Coefficients

Author: YuanYuan
Tutor: LiHuiLai
School: Jilin University
Course: Basic mathematics
Keywords: The L ~~ 1 coefficient Nonlinear Existence
CLC: O175.2
Type: Master's thesis
Year: 2007
Downloads: 3
Quote: 0
Read: Download Dissertation


In this paper,we establish the existence of solutions for a class of nonlinear equations with L1 coefficients. This equation has applications in the consideration of electrodiffusion in thin-film conductors. In the study of thin-film conductors,ion diffusion is an important phenomenon as electrodiffusion degradation can lead to metallization failure. Electrodiffusion arise under many circumstances,including electromechanical stresses due to physicotechnological conditions of thin-film deposition and the physicochemical properties of the film and substrate materials.Considering a thin-film of finite length with endpoints at x = 0 and x = 1 which is one-dimensional case of the parabolic equation. Then,under electrodiffusion,the electromechanical stress u = u(x, t) can be modeled as follows: where 5(x) is the one-side Dirac delta function,and for any continuous functionf(x,t)is a given function depending on the electromechanical stress of thin-film deposition and we discuss the initial and boundary conditionsu(x,0) =φ(x),0≤x≤1,u(0,t) = u0(t),u(1,t) = u1(t),0≤t≤T.For the problem, we are interested in the Diracδ-function, as coefficients are measure functions. In fact, the studies of parabolic equations are always restricted to the complexion that the initial value is measure function,and the measure as coefficients is not familiar.The equations we considere in this paper are more common u(0,t) = u0(t) ,u(1,t) = u1(t) ,0≤t≤T , (1.2)u(x,0) =φ(x) ,0≤x≤1, (1.3)hereμ∈L1 ,A is a nonlinear function of u and A∈C1, A’≥ε00 is a positive constant. The failure of principle of superposition made the problem more complex. For solving the problem, we discussed existence of solutions for a class of ordinary differential equations first and got a family comparison functions. Then we consider the existence of the problem (1.1)-(1.3).In section 2,we discuss the existence of solutions of two ordinary differential equations and displayed the depended relation that the solutions on L1 coefficients f(x) and g(x).For the initial value problem of ordinary differential equation as follows(A(u(x)))" = f(x)(u(x))’ + g(x), (2.1)u(0)=u0, u’(0) = u1, (2.2)here f(x),g(x)∈L1[0,1]If we denote v = A(u) ,u = A-1(v) = a(v) ,we could get a equation which is equivalent to (2.1)(2.2)v"(x) = f(x)(a(v(x)))’ + g(x), (2.3)v(0) = A(u0), v’(0) = A’(u0)u1. (2.4)We assume thatH1:A∈C1,A’>0,a = A-1;H2: a, a’ satisfy Lipschitz condition ,Lipschitz constant is L.Theorem 2.1 If condition H2 is satisfied,there exists a solution v∈W2,1 [0,1] of the equations(2.3)(2.4).If we use (2.4) and integrate (2.3) we get For some 0 <δ< 1 which satisfied a certain condition, there exists a solution u∈C1[x0,x0+δ] of (2.3)(2.4) andδcan be expended to 1. So the equations (2.3)(2.4) have solution v∈C1[0, 1].Then we give the uniform bound of the solution.Proposition 2.1 If conditions of Theorem 2.1 is satisfied,there exists a constant C depend only on ||f||L1, ||g||l1 and initial value,such that||v||C1[0,1]≤C.In section 3,we come back to discussed existence of solutions of the parabolic problem.We consider the following regularized problemuε(0, t) =uε0(t) , uε(1, t)=uε1(t),0≤t≤T, (3.2)uε(x,0) =φε(x) ,0≤x≤1 . (3.3)We assumeH1: A∈C1 and there existsε0 > 0 such that A’(s)≥ε0, denote a = A-1 ,a and a’ satisfy Lipschitz condition with Lipschitz constant L;H2:μ∈L1 is a smooth function ,andμ≥0 ,μ’ < 0;H3ε∈C1 uniform approximatesφand <φε,φ’εis uniform bounded;H4: ,uε0,uε1∈C1 uniform approximate u0,u1;H5: h(x, t)∈C1(Q) ,and there exist h1(x), h2(x)∈C1[0, 1] ,such that -h1(x) <-h(x,t) <-h2(x).There is a smooth solution uεof the problem, whose existence follows from the classical theory.Using the results we get from the ordinary equation we have discussed,from comparison principle,we get Lemma 3.1 Let uεbe a solution of problem (3.1)-(3.3) then|uε(x,t)|≤M, (3.4)where the constant M depend only on ||/||L1 , ||h||Land initial value.According to the classical theory of parabolic equations we have prior estimate as followsLemma 3.2 Let uεbe a solution of the problem(3.1)-(3.3), thenwhere the constant C depend only on ||f||L1 , ||h||land initial value.Lemma 3.3 Let uεbe a solution of the problem(3.1)-(3.3), thenwhere the constant C depend only on ||f||L1 , ||h||land initial value. Using Lemma 3.2 and 3.3, under the assumption of A,we haveLemma 3.4 Let uεbe a solution of the problem(3.1)-(3.3), thenwhere the constant C depend only on ||f||L1 ,‖h‖Land initial value.Proposition3.1 Let uεbe a solution of the problem(3.1)-(3.3), then for any (x1,t1),(x2,t2)∈Q,|A(uε(x1,t1)) - A(uε(x2,t2))|≤C(|x1 - x2| + |t1 - t2|1/2), where the constant C depend only on ||f||L1 , ||h||land initial value. Proposition 3.2 Let uεbe a solution of the problem(3.1)-(3.3), then for any (x1,t1),(x2,t2)∈Q,|uε(x1,t1) - uε(x2, t2)|≤C(|x1 - x2| + |t1 - t2|1/2),α∈(0,1),where the constant C depend only on ||f||L1 , ||h||land initial value.We assumeμ, is bounded regularized Borel measure,then there exists a smooth function sequence,such that for any continuous functionψ(x),and‖μεL1≤C, where dμ* =μdx.At last we give the definition of generalized solutions and get the existence of the solution.Definition3.1 A function u∈C1,1/2(Q) is called a generalized solution of the boundary value problem(1.1)-(1.3),if the integral equalityis fulfilled for any functionψ∈C(Q) withψ(0,t) =ψ(1,t) =ψ(x,T) = 0.Theorem3.1 If conditions H1—H5 are satisfied, then the first boundary value problem (1.1)-(1.3) admits solutions.As above ,we complete the proof of the existence and uniqueness of solutions of the problem (1.1)-(1.3).

Related Dissertations

  1. Satellite Attitude Determination Based on Gyro and Star Sensor,V448.2
  2. Analysis and Study of Abutment Stability in Concrete High Arch Dam by Three-Dimensional Nonlinear Finite Element Method,TV642.4
  3. Infrared Imaging Target Simulation Resistance Array Control System Development,TJ765.4
  4. Research on Anti-periodic Solutions of Delayed Cellular Neural Networks without Assuming Global Lipschitz Conditions,TP183
  5. Research on Adaptive Control of Recovery of UUV with Near Wall Constraint,TP273
  6. Integrable Discretizations to a Class of Soliton Equations,O175.2
  7. Experimental Study on Heavy Metals Adsorption or Release from Sediments of East Dongting Lake,X524
  8. Living conditions of migrant workers and its influencing factors,D412.6
  9. Rate-dependent Endochronic Damage Constitutive Model for Concrete,TU528
  10. Existence of Vocational School Class Teachers,G715.1
  11. Investigations on Related Problems of the Integrable Coupling Systems and the Exact Solutions of Nonlinear Differential-Difference Equations,O175.7
  12. Optical Properties of Chlorphyll-a in Reservoir Water and Its Concentration Inversion Models by Remote Sensing,S127
  13. Coal and gas outburst Hou Wasi Dynamic Emission Law,TD713
  14. Zhaoguan Lower Coal Group water inrush prediction and control techniques,TD745
  15. \u00C2\u00BF \u00BE ??C ?? CY \u00BF??\u00A5\u00A5 C?? \u00AF \u0153\u00E6 ?? '\u00E6 ? ?? ?? S11 \u301C,TD745
  16. Frobenius nonlinear evolution equations integrable solutions and the expansion of integrable systems,O175.5
  17. Qualitative Study of Almost Periodic Solutions of Some Competition System,O175
  18. Research on China’s Death Penalty System,D924.1
  19. The Research and Application of Modified Algorithms About Fuzzy Predictive Functional Control,TP273
  20. Problems and Countermeasures of Medium-city TV,G229.24
  21. The Existence of Global Solution for Stochastic Functional Differential Equation,O211.63

CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations
© 2012 www.DissertationTopic.Net  Mobile