An exhaust system, comprising: a trapping filter that is disposed at an exhaust line to trap particulate material of exhaust gas; a nitrogen oxide purification catalyst that is disposed at a downstream side of the trapping filter to eliminate nitrogen oxide of the exhaust gas; an injector that selectively injects fuel in the exhaust line; a fuel cracking catalyst that is disposed between the injector and the trapping filter at the exhaust line, transforms the injected fuel to a reducing agent, and raises the temperature of the downstream side thereof; wherein the nitrogen oxide purification catalyst uses the reducing agent that is formed by the fuel cracking catalyst to detach and reduce the nitrogen oxide in the nitrogen oxide purification catalyst; and a control portion that controls the injector to inject a predetermined amount of the fuel for a predetermined time period so as to raise the temperature of the trapping filter to a predetermined regeneration temperature and eliminate the particulate material trapped in the trapping filter, wherein the control portion detects a degradation rate of the fuel cracking catalyst, and varies the predetermined regeneration temperature of the trapping filter depending on the degradation rate, and characterized in that the control portion compensates the predetermined injection amount and the predetermined time period of the injector when a temperature difference between an upstream side and a downstream side of the fuel cracking catalyst is larger than a predetermined value, a temperature difference between an upstream side and a downstream side of the nitrogen oxide purification catalyst is smaller than a predetermined value, an oxygen concentration difference between an upstream side and the downstream side of the fuel cracking catalyst is larger than a predetermined value, and a purification ratio of the nitrogen oxide purification catalyst is larger than or equal to a predetermined value. a trapping filter that is disposed at an exhaust line to trap particulate material of exhaust gas; a nitrogen oxide purification catalyst that is disposed at a downstream side of the trapping filter to eliminate nitrogen oxide of the exhaust gas; an injector that selectively injects fuel in the exhaust line; a fuel cracking catalyst that is disposed between the injector and the trapping filter at the exhaust line, transforms the injected fuel to a reducing agent, and raises the temperature of the downstream side thereof; wherein the nitrogen oxide purification catalyst uses the reducing agent that is formed by the fuel cracking catalyst to detach and reduce the nitrogen oxide in the nitrogen oxide purification catalyst; and a control portion that controls the injector to inject a predetermined amount of the fuel for a predetermined time period so as to raise the temperature of the trapping filter to a predetermined regeneration temperature and eliminate the particulate material trapped in the trapping filter, wherein the control portion detects a degradation rate of the fuel cracking catalyst, and varies the predetermined regeneration temperature of the trapping filter depending on the degradation rate, and characterized in that the control portion compensates the predetermined injection amount and the predetermined time period of the injector when a temperature difference between an upstream side and a downstream side of the fuel cracking catalyst is larger than a predetermined value, a temperature difference between an upstream side and a downstream side of the nitrogen oxide purification catalyst is smaller than a predetermined value, an oxygen concentration difference between an upstream side and the downstream side of the fuel cracking catalyst is larger than a predetermined value, and a purification ratio of the nitrogen oxide purification catalyst is larger than or equal to a predetermined value.