The inverse problem for a quasilinear integro-differential equation
The author deals with the problem of recovering function \(\varphi\) in the quasi-linear integro-differential equation \[ \begin{multlined} D_x^2u(x,t) - cD_xu(x,t) - d\varphi(u(x,t)) + \int_0^t B(x,t,s)\varphi((u(x,s)) ds = 0,\\ (x,t)\in Q_T=(0,l)\times (0,T),\end{multlined} \tag{1} \] subject to the boundary conditions \[ u(0,t)=\mu(t),\qquad u(l,t) + hD_xu(l,t) = 0,\qquad 0 < t < T, \tag{2} \] and to the additional information \[ D_xu(0,t)=g(t),\qquad 0 < t < T. \tag{3} \] The author assumes that \(c\), \(d\), and \(h\) are positive constants, while the functions \(B\), \(\mu\), and \(g\) satisfy the conditions: i) \(B,D_tB\in C(D)\), where \(D=\{(x,t,s): 0<x<1\), \(0\leq s\leq t\leq T\}\); ii) \(\mu\in C^3([0,T])\), \(\mu(0)= 0\), \(\mu'(t)> 0\), \(t\in [0,T]\); iii) \(g\in C^1([0,T])\), \(g(t)<0\) for any \(t\in [0,T]\). For a (classical) solution to the inverse problem the author means a pair \((u,\varphi)\) satisfying: iv) \(u\in C^1(Q_T)\) such that \(D_tD_xu,D^2_xu\in C(Q_T)\); v) \(\varphi \in C^1({\mathbb R})\cap C^3([0,\mu(T)])\); \(\varphi(0)=0\), \(\varphi'(s)> 0\), \(s\in {\mathbb R}\). The aim of the paper consists in showing the uniqueness of the solution \((u,\varphi)\) to problem (1)--(3). Under suitable sign assumptions involving the kernel \(B\), its derivative \(D_tB\) and \(\int_0^t D^j_tB(x,t,s) ds\), \(j=0,1\), the author can show, by suitable maximum principle procedures, that the derivatives \(D_xu\) and \(D_tu\) are, respectively, strictly negative and positive in \(Q_T\). Taking advantage of this result, he can prove that, if \((u_i,\varphi_i)\), \(i=1,2,\) solve problem (1)--(3), then the derivatives of \(\varphi_1\) and \(\varphi_2\) up to the third order coincide at \(t=0\). Finally, the uniqueness of the inverse problem is ensured when two functionals \(k_0\) and \(I_0\) related to the following initial value problem satisfy the condition \(|k_0|+I_0<1\): \[ z''(x) - [c^2/4 + d\varphi'(0)]z(x) = 0,\quad 0 < x < l,\qquad z(l)=1,\quad z'(l)= -H, \] where \(\varphi'(0):=\varphi_1'(0)=\varphi_2'(0)\) and \(H = (2 + hc)/(2h)\).
quasilinear integrodifferential equations
scientific article; zbMATH DE number 1915949