The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for u t = div(|∇u|. By constructing suitable function spaces and applying the method of Galerkin's approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable test-function, make integral estimates, and apply Gronwall's inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that p t (x, t) is only negative and integrable, which is weaker than most of the other papers required.
The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equationDue to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent α(x), some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of α(x).
The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation −div( ( )∇ ) = ( )/ ( ) with ∈ (Ω) ( ⩾ 1) and ( ) > 0. The results show the dependence of the summability of in some Lebesgue spaces and on the values of ( ).
<p style='text-indent:20px;'>In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.</p>
In this paper, we study an initial boundary value problem for a p-Laplacian hyperbolic equation with logarithmic nonlinearity. By combining the modified potential well method with the Galerkin method, the existence of the global weak solution is studied, and the polynomial and exponential decay estimation under certain conditions are also given. Moreover, by using the concavity method and other techniques, we obtain the blow up results at finite time.
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