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We use Cisinski's machinery to construct and study model structures on the category of simplicial sets whose classes of fibrant objects generalize quasicategories. We identify a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects and show that there is a model structure whose fibrant objects are precisely those which satisfy this condition. We also identify a localization of this model structure whose fibrant objects satisfy a "special horn lifting" condition similar to the one characterizing quasi-categories. This special horn model structure leads to a conjecture characterization of the bijective-on-0-simplices trivial cofibrations of the Joyal model structure. We also discuss how these model structures all relate to one another and to the minimal model structure.

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We present a method for the solution of the Cauchy problem for three broad classes of nonlinear parabolic equations UDC 517.43 and OU (t, x) bU (t, x) = f(t, ALU(t, x)with the infinite-dimensional Laplacian A L.During the entire 20 th century, the theory of linear equations with infinite-dimensional Laplacians (LEvy Laplacians [1]) has attracted the attention of many mathematicians. This attention is stimulated by interesting properties of the LEvy Laplacians (which often do not have finite-dimensional analogs) and by important applications (e.g., the Yang-Mills and Laplace-L&y equations are equivalent [2]).At the same time, the formation of the theory of quasilinear and nonlinear equations with LEvy Laplacians is at the beginning stage. The investigation of these equations was carded out only in the works of LEvy [1] (the Dirichlet problem), Shilov [3] (the Dirichlet problem and mixed problem), and Feller [4-6] (the Dirichlet problem and Riquier problem). Quasilinear equations were studied by Sokolovskii in [7].The present paper is devoted to the solution of the Cauchy problem for nonlinear parabolic equations with LEvy Laplacians. We consider the following three classes of nonlinear equations: bU(t, x) -f(t, ALU(t,x)) (f(t,~) isafunctionon [O,T]• 3t OU(t, x) -f(U(t, x), ALU(t, x)) (f(~, ;) is a function on R2). 3tThe solutions of the Cauchy problem for nonlinear equations are considered in functional classes for which there exists a solution of the Cauchy problem for the "heat equation" OV(t, x) = ALV(t,x ). btThis is explained by the fact that, in many works, the solutions of the Cauchy problem for linear equations with LEvy Laplacians were obtained in various functional classes (see the bibliography in [8]).UkrNIMOD, Kiev.

We prove that every 2-Segal space is unital.