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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
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Abstract


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).

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations
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