The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in whic...The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.展开更多
The homogenization of the nonlinear degenerate parabolic equations, diva(x/ε,t/ε,u,u)=f(x,t) is studied, where a(y,t,μ,λ) is periodic in (y,t) and b may be a nonlinear function whose prototype is |μ|~r sign u wi...The homogenization of the nonlinear degenerate parabolic equations, diva(x/ε,t/ε,u,u)=f(x,t) is studied, where a(y,t,μ,λ) is periodic in (y,t) and b may be a nonlinear function whose prototype is |μ|~r sign u with r>0.展开更多
In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is...In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11061018,11261029)the Youth Foundation of Lanzhou Jiaotong University(No.2011028)+1 种基金the Long Yuan Young Creative Talents Support Program(No.252003)the Joint Funds of the Gansu Provincial Natural Science Foundation of China(No.1212RJZA043)
文摘The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.
基金supported by the National Natural Sciences Foundation of China (No.19701018).
文摘The homogenization of the nonlinear degenerate parabolic equations, diva(x/ε,t/ε,u,u)=f(x,t) is studied, where a(y,t,μ,λ) is periodic in (y,t) and b may be a nonlinear function whose prototype is |μ|~r sign u with r>0.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 10631020)Basic Research Grant of Tsinghua University (Grant No. JCJC2005071).
文摘In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.
