Description: We study the periodic Dirichlet problem for a semilinear wave equation with discontinuous nonlinearity. First we establish a continuation theorem for a semilinear operator equation in a Hilbert space, where a key tool is the Berkovits-Tienari degree theory for a class of perturbations of monotone type of a densely defined closed linear operator. Applying the continuation theorem, we prove the main results on the solvability of the given semilinear wave equation, with the aid of spectral theory for densely defined closed linear operators.MSC: 35L05, 35R05, 47H04, 47H05, 47H11.

Description: The presence of aerosol has resulted in serious limitations in the data coverage and large uncertainties in retrieving carbon dioxide (CO2) amounts from satellite measurements. For this reason, an aerosol retrieval algorithm was developed for the Thermal and Near-infrared Sensor for carbon Observation-Cloud and Aerosol Imager (TANSO-CAI) launched in January 2009 on board the Greenhouse Gases Observing Satellite (GOSAT). The algorithm retrieves aerosol optical depth (AOD), aerosol size information, and aerosol type in 0.1° grid resolution by look-up tables constructed using inversion products from Aerosol Robotic NETwork (AERONET) sun-photometer observation over Northeast Asia as a priori information. To improve the accuracy of the TANSO-CAI aerosol algorithm, we consider both seasonal and annual estimated radiometric degradation factors of TANSO-CAI in this study. Surface reflectance is determined by the same 23-path composite method of Rayleigh and gas corrected reflectance to avoid the stripes of each band. To distinguish aerosol absorptivity, reflectance difference test between ultraviolet (band 1) and visible (band 2) wavelengths depending on AODs was used. To remove clouds in aerosol retrieval, the normalized difference vegetation index and ratio of reflectance between band 2 (0.674 μm) and band 3 (0.870 μm) threshold tests have been applied. To mask turbid water over ocean, a threshold test for the estimated surface reflectance at band 2 was also introduced. The TANSO-CAI aerosol algorithm provides aerosol properties such as AOD, size information and aerosol types from June 2009 to December 2013 in this study. Here, we focused on the algorithm improvement for AOD retrievals and their validation in this study. The retrieved AODs were compared with those from AERONET and the Aqua/MODerate resolution Imaging Sensor (MODIS) Collection 6 Level 2 dataset over land and ocean. Comparisons of AODs between AERONET and TANSO-CAI over Northeast Asia showed good agreement with correlation coefficient (R) 0.739 ± 0.046, root mean square error (RMSE) 0.232 ± 0.047, and linear regression line slope 0.960 ± 0.083 for the entire period. Over ocean, the comparisons between Aqua/MODIS and TANSO-CAI for the same period over Northeast Asia showed improved consistency, with correlation coefficient 0.830 ± 0.047, RMSE 0.140 ± 0.019, and linear regression line slope 1.226 ± 0.063 for the entire period. Over land, however, the comparisons between Aqua/MODIS and TANSO-CAI show relatively lower correlation (approximate R = 0.67, RMSE = 0.40, slope = 0.77) than those over ocean. In order to improve accuracy in retrieving CO2 amounts, the retrieved aerosol properties in this study have been provided as input for CO2 retrieval with GOSAT TANSO-Fourier Transform Spectrometer measurements.