| Keywords [eng] |
Sobolevo erdvės, silpnas sprendinys, labai silpnas sprendinys, šilumos lygtis, netiesinė sąlyga. Sobolev spaces, Weak solution, Very weak solution, Heat problem, Nonlinear condition. |
| Abstract [eng] |
In this thesis, an inverse heat conduction problem is studied, where functions $E$ and $u_0$ are prescribed and the goal is to find a pair $(u, f)$, satisfying the differential equation $u_t(x, t) - \Delta u(x, t) = f(x, t)$, initial condition $u(x, 0) = u_0(x)$, the Dirichlet boundary condition $\left.u \right|_{\partial \Om \times [0, T]} = 0$ as well as an additional nonlinear nonlocal condition $\int_\Om |u(x, t)|^2 dx = E(t)$ for all $t \in [0, T]$. When $E \in W^{1,2}(0, T)$, we formulate the definition of a weak solution for this problem and prove that there exists at least one such solution. In the case when $E$ is only from the space $L^2(0, T)$, we formulate the definition of a very weak solution and prove that there exists at least one such solution. |
