The Inverse Coefficient Problem For A Hyperbolic Equation With A Point Source And The Null Controllability And Inverse Coefficient Problem For A Singular Heat Equation With Variable Coefficients

In the first chapter of this paper,the subject of inverse problem of PDEs is intro-duced,and several kinds of inverse problems are given.In the second chapter of the paper,for the solution to I (?)t2u(x,t)-?u(x,t)+q(x)u(x,t)=8(x,t)u|t<0=0,we consider an inverse problem of determining q(x),x?? from data f=|sT and g=((?)u/(?)n)|sT Here ?(?){(X1,x2,x3)?R3|x1>0}is a bounded domain,ST={(x,t)|x?(?)?,|x|<t<T+|x|},n=n(x)is the outward unit normal n to (?)?,and T>0.For suitable T>0,prove a Lipschitz stability estimation:?q1-q2?L2(?)?C{?f1-f2?H1(ST)+?g1-g2?L2(ST)},provided that q1 satisfies a priori uniform boundedness conditions and satisfies a priori uniform smallness conditions,where uk is the solution to problem(2.1)with q=qk,fk and gk are boundary observation data corresponding to uk,respectively,k=1,2.In the third chapter of the paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential:(?)tu(x,t)—div(p(x)(?)u(x,t))-|2|u(x,t)=f(x,t),0??,0<t<T.Here ? is a positive real constant.It was proved in the paper of Goldstein and Zhang(2003)that the equation is well-posedness when 0???P1(n—2)2/4,and in this paper,we mainly consider the case 0??<(p12/p2)(n-2)2/4,where P1,P2 are two positive constants which satisfy:0<P1?p(x)?p2,Vx??.We extend the specific Carleman estimates in the paper of Ervedoza(2008)and Vancostenoble(2011)to the equation we consider and apply it to deduce an observability inequality for the system.By this inequality and the classical HUM method,we obtain that we can control the equation from any non-empty open subset as for the heat equation.Moreover,we will study the case ?>p2(n-2)2/4.We consider a sequence of regularized potenticrls ?l(|x|2+?2),and prove that we cannot stabilize the corresponding systems uniformly with respect to?>0,due to the presence of explosive modes which concentrate around the singularity.In the fourth chapter of the paper are the uniqueness and stability of the inverse co-efficient problem for the system:(?)1u(x,t)-(?)·(P(x)Vu)-|x|2/?u(x,t)=0,(x,t)??×(0,T),where 0?? is a bounded domain,P(x)?B={P(x)?C3(?);M1?P(x)?M2,?(?)P?C(?))?M3)is unknown.This is an inverse problem of determining P(x),x??\{0},from observation data u(x,t),(x,t)??×(t0,T)and (?).(P(?)u(x,T0))+|x|2/??u(x,T0),x??·Here,T0=(t0+T)/2,for (?)0<t0<T,0(?)?is an arbitrary non-empty open subset of ?.In this article,we prove a local estimate for the inverse problem by a method on the basis of Carleman estimates for inverse coefficient prob-lems with a finite number of measurements.The proof is done similarly to Section 6 in Yamamoto M.[61],in which inverse problems for classical parabolic equations without any singular inverse-square potential are considered.