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In [8] Brézis and Friedman prove that certain nonlinear parabolic equations, with the δ-measure as initial data, have no solution. However in [9] Colombeau and Langlais prove that these equations have a unique solution even if the δ-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais their result proving that we may take any generalized function as the initial data. Our approach relies on resent algebraic and topological developments of the theory of Colombeau generalized functions and results from [1]. * 2000 Mathematics Subject Classification: Primary 46F30 Secondary 46T20

Abstract-For large volume visualization, an image-based quality metric is difficult to incorporate for level-of-detail selection and rendering without sacrificing the interactivity. This is because it is usually time-consuming to update view-dependent information as well as adjust to transfer function changes. In this paper, we introduce an image-based level-of-detail selection algorithm for interactive visualization of large volumetric data. The design of our quality metric is based on an efficient way to evaluate the contribution of multiresolution data blocks to the final image. To ensure real-time update of the quality metric and interactive level-of-detail decisions, we propose a summary table scheme in response to run-time transfer function changes, and a GPU-based solution for visibility estimation. Experimental results on large scientific and medical data sets demonstrate the effectiveness and efficiency of our algorithm.