Global Strong Solution With BV Derivatives to Singular Solid-on-Solid model With Exponential Nonlinearity

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t=\mathrm{\nabla }\cdot (\frac{1}{\mathrm{\mid }\mathrm{\nabla }h\mathrm{\mid }}\mathrm{\nabla }{e}^{\frac{\delta E}{\delta h}})=\mathrm{\nabla }\cdot (\frac{1}{\mathrm{\mid }\mathrm{\nabla }h\mathrm{\mid }}\mathrm{\nabla }{e}^{-\mathrm{\nabla }\cdot (\frac{\mathrm{\nabla }h}{\mathrm{\mid }\mathrm{\nabla }h\mathrm{\mid }})})$$ where total energy $E=\int |\nabla h|$isthetotalvariationof$h$\mathrm{.}Usingalogarithmiccorrection$E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity $h_{xx}^{+}$ happens.